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CK-12 Foundation CK-12 Texas Instruments Geometry Teacher's Edition Say Thanks to the Authors Click http://www.ckl2.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive content, visit www.ckl2.org CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook mate- rials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FlexBook®, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform®. Copyright © 2011 CK-12 Foundation, www.ckl2.org The names "CK-12" and "CK12" and associated logos and the terms "FlexBook®", and "FlexBook Platform®", (collectively "CK-12 Marks") are trademarks and service marks of CK-12 Foundation and are protected by federal, state and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ckl2.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution/Non-Commercial/Share Alike 3.0 Un- ported (CC-by-NC-SA) License (http://creativecommons.0rg/licenses/by-nc-sa/3.O/), as amended and updated by Creative Commons from time to time (the "CC License"), which is incorporated herein by this reference. Complete terms can be found at http://www.ckl2.org/terms. Printed: August 2, 2011 /lexboo< next generation textbtwks Editor Lori Jordan www.ckl2.org Contents 1 TE Introduction to Geometry - TI 1 1.1 Geometry TI Resources Flexbook 1 2 TE Basics of Geometry - TI 2 2.1 Midpoints in the Coordinate Plane 2 2.2 Vertical and Adjacent Angles 5 3 TE Reasoning and Proof - TI 8 3.1 Conditional Statements 8 4 TE Parallel and Perpendiculuar Lines - TI 12 4.1 Parallel Lines cut by a Transversal 12 4.2 Transversals 13 4.3 Perpendicular Slopes 15 5 TE Congruent Triangles - TI 17 5.1 Interior and Exterior Angles of a Triangle 17 5.2 Congruent Triangles 20 5.3 Triangle Sides and Angles 24 6 TE Relationships within Triangles - TI 27 6.1 Perpendicular Bisector 27 6.2 Hanging with the Incenter 29 6.3 Balancing Point 31 6.4 Hey Ortho! What's your Altitude? 33 7 TE Quadrilaterals - TI 37 7.1 Properties of Parallelograms 37 www.ckl2.org 11 7.2 Properties of Trapezoids and Isosceles Trapezoids 39 7.3 Properties of Rhombi, Kites and Trapezoids 40 8 TE Similarity - TI 43 8.1 Constructing Similar Triangles 43 8.2 Side-Splitter Theorem 46 8.3 Perspective Drawing 48 9 TE Right Triangle Trigonometry - TI 52 9.1 The Pythagorean Theorem 52 9.2 Investigating Special Triangles 56 9.3 Ratios of Right Triangles 59 10 TE Circles - TI 63 10.1 Chords and Circles 63 10.2 Inscribed Angles Theorem 65 10.3 Circle Product Theorems 67 11 TE Perimeter and Area - TI 71 11.1 Diameter and Circumference of a Circle 71 11.2 From the Center of the Polygon 73 12 TE Surface Area and Volume - TI 76 12.1 Surface Area of a Cylinder 76 13 TE Transformations - TI 79 13.1 Transformations with Lists 79 13.2 Reflections and Rotations 82 13.3 Perspective Drawing 84 111 www.ckl2.org www.ckl2.org IV Chapter 1 TE Introduction to Geometry TI 1.1 Geometry TI Resources Flexbook Teacher's Edition Introduction This flexbook contains Texas Instruments (TI) Resources for the TI-83, TI-83 Plus, TI-84, and TI-84 SE. All the activities in this flexbook supplement the lessons in the student edition. Teachers may need to download programs from www.timath.com that will implement or assist in the activities. Each activity included is designed to help the teacher with each activity in the TI Resources Flexbook, Student Edition. All activities are listed in the same order as the Student Edition. There are also corresponding links in the 1st Edition of Geometry, 2nd Edition, and Basic Geometry. • Geometry, first edition: http://www.ckl2.org/flexr/flexbook/805 • Geometry, second edition: http://www.ckl2.org/flexr/flexbook/3461 • Basic Geometry: http://www.ckl2.org/flexr/flexbook/3129 Any activity that requires a calculator file or application, go to http://www.education.ti.com/calculators/ downloads and type the name of the activity or program in the search box. www.ckl2.org Chapter 2 TE Basics of Geometry - TI 2.1 Midpoints in the Coordinate Plane This activity is intended to supplement Geometry, Chapter 1, Lesson 4- ID: 8614 Time required: 40 minutes Topic: Points, Lines &: Planes • Given the coordinates of the ends of a line segment, write the coordinates of its midpoint. Activity Overview In this activity, students will explore midpoints in the coordinate plane. Beginning with horizontal or vertical segments, students will show the coordinates of the endpoints and make a conjecture about the coordinates of the midpoint. This conclusion is extended to other segments in the coordinate plane. Teacher Preparation • This activity is designed to be used in a high school or middle school geometry classroom. • The screenshots on pages 1-3 demonstrate expected student results. • The Coordinate Midpoint formula for the midpoint of {x\,y\) and (x2,y2) is \ 2 2 , ^ 2 >2 )• This can also be expressed as "The coordinates of the midpoint of a line segment are the averages of the coordinates of the endpoints." Classroom Management • This activity is designed to be student- centered with the teacher acting as a facilitator while students work cooperatively. The student worksheet provides a place for students to record their observations. • Depending on student skill level, you may wish to use points with integer coordinates, or only positive values. • Note: The coordinates can display 0, 1, or 2 decimal digits. If digits are displayed, the value shown will round from the actual value. To ensure that a point is actually at an integer value rather than a rounded decimal value, do the following: www.ckl2.org 2 1. Move the cursor over the coordinate value so it is highlighted. 2. Press + to display additional decimal digits or - to hide digits. • The ideas contained in the following pages are intended to provide a framework as to how the activity will progress. • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. Associated Materials • Student Worksheet: Midpoints in the Coordinate Plane http://www.ckl2.org/flexr/chapter/ 9686 • Cabri Jr. Application Problem 1 — Midpoints of Horizontal or Vertical Segments Step 1: • Students should open a new Cabri Jr. file. If the axes are not currently showing, they should select Hide/Show > Axes. They will construct a horizontal segment in the first quadrant using the Segment tool. Step 2: • Students will select Coord. & Eq. and show the coordinates for the endpoints of the segment. If the coordinates of the endpoints are not integers, they need to use the Hand tool to drag the endpoints until the coordinates are integers. Step 3: Students should now make a prediction about the coordinates for the midpoint of the segment. To check their predictions, students will select Midpoint, construct the midpoint of the segment, and then show its coordinates. —¥ m m m D D P Ci,i> CH/i) Step 4: Before moving on, students need to hide the coordinates of the midpoint with the Hide/Show > Object tool. They should use the Hand tool to drag the segment to another location. If you drag the entire segment, it will remain horizontal. 3 www.ckl2.org Students can make a prediction about the new coordinates of the midpoint and check their prediction by showing the coordinates of the midpoint. Step 5: • Repeat this exploration with a new segment. Use a vertical segment. If desired, have students explore midpoints of segments whose endpoints do not have integer coordinates, or are not in Quadrant 1. Problem 2 — Midpoints of Diagonal Segments Step 1: • Instruct students to open a new Cabri Jr. file. If needed, select Hide/Show > Axes to show the coordinate axes. • They should begin by using the Segment tool to construct a diagonal segment in the first quadrant. Step 2: • Students need to select Coord. & Eq. and show the coordinates for the endpoints of the segment. If the coordinates of the endpoints are not integers, they should use the Hand tool to drag the endpoints to make the coordinates integers. Step 3: • Students should now make a prediction about the coordinates for the midpoint of the segment. To check their predictions, students will construct the midpoint of the segment and show the coordinates of the midpoint. Step 4: • Students should now hide the coordinates of the midpoint with the Hide/Show > Object tool. • Using the Hand Tool, students should drag the segment to another location. If the entire segment is selected, it will keep the same diagonal slant. Students will make a prediction about the new coordinates of the midpoint and check their prediction by showing the coordinates of the midpoint. Step 5: www.ckl2.org 4 • Repeat this exploration with a new segment. If desired, direct students to again consider segments whose endpoints do not have integer coordinates or are not in Quadrant 1. Step 6: • In pairs, students should discuss the following topic: Describe in words how to find the coordinates of the midpoint of a segment if you know the coordinates of the endpoints. Try to write a formula or a rule for midpoints. 2.2 Vertical and Adjacent Angles This activity is intended to supplement Geometry, Chapter 1, Lesson 5. ID: 10893 Time required: 15 minutes Activity Overview In this introductory or review activity, students will explore vertical and adjacent angles. They will define and identify pairs of angles. Then they will change the intersecting lines of a geometric model to make conjectures about the relationships of the pairs of angles. Topic: Points, Lines & Planes • Congruency of vertical angles • Adjacent angles formed by two intersecting lines are supplementary. Teacher Preparation and Notes • This activity was written to be explored with Cabri Jr. • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. • To download the calculator file, go to http: //www . education, ti . com/calculators/downloads/ US/ 'Activities/Detail? id=10893 and select VERTICAL. 8xv. Associated Materials • Student Worksheet: Vertical and Adjacent Angles http://www.ckl2.org/flexr/chapter/9686, scroll down to the second activity. • Cabri Jr. Application . VERTICAL. 8xv Problem 1 — Exploring Vertical Angles Students should first define the term vertical angles using their textbook or other source. After opening Cabri Jr., student should press Y = and select Open and then VERTICAL to view the file. Introduce the geometric model (two intersecting lines). 5 www.ckl2.org Students should name the two pairs of vertical angles of the model. Direct students to explore the model independently, by grabbing and dragging points B and/or C. To grab a point, move the cursor over the point and then press a. The cursor will change to a closed fist. Press a again to release the point. Then they need to answer the questions on the worksheet. MHQE: 150* HCDEi 150* HE:Q>: £0* MADEi SO* ~"' A M^ --" Problem 2 — Exploring Adjacent Angles Students are to repeat the steps from Problem 1 with adjacent angles. They will need to first define the term adjacent angles using a textbook or other source. Students are to use the same geometric model of intersecting lines from Problem 1. They need to name the four pairs of adjacent angles. Have them explore the model independently and make a conjecture about adjacent angles. They can also use the Calculate tool from the F5 menu to add the pairs of angles confirming that adjacent angles are supplementary. To do this, select one angle measurement, press +, and then select the adjacent angle measurement. Solutions 1. Two angles whose sides are opposite rays. 2. IAOB and ICOD; IBOC and lAOD 3. Sample answers. Table 2.1: Location I s -ynd ord Ath mlAOB 130.6 mlBOC 49.4 mlCOD 130.6 mlAOD 49.4 118.5 90.4 79.4 61.5 89.6 110.6 118.5 90.4 79.4 61.5 89.6 100.6 4. If lAOD and IBOC are vertical angles, then the mlAOD = mlBOC. 5. If lAOB and ICOD are vertical angles, then the mlAOB = mlCOD. 6. Vertical angles are congruent. 7. Adjacent angles are two coplanar angles that have a common side and a common vertex but no common interior points. www.ckl2.org 6 8. IAOB and IBOC; IBOC and ICOD; LAOD and LCOD; LAOD and lAOB 9. Adjacent angles formed by two intersecting lines are supplementary. 10. If lAOB andlBOC are adjacent angles formed by two intersecting lines, then LAOB and IBOC are supplementary. 11. 3x = 75 x = 25 y = 180 - 75 = 105 12. x + 10 = 4x - 35 3x = 45 x= 15 www.ckl2.org Chapter 3 TE Reasoning and Proof - TI 3.1 Conditional Statements This activity is intended to supplement Geometry, Chapter 2, Lesson 2. ID: 8746 Time required: 40 minutes Activity Overview In this activity, students will write logical statements related to the given conditional statement. They will explore whether the statements are true or false and find counterexamples for false statements. These explo- rations will involve the slopes of parallel and perpendicular lines and lengths of collinear and noncollinear segments. Topic: Inductive & Deductive Reasoning • Write the inverse, converse, and contrapositive statements corresponding to a given conditional state- ment. • Use a counterexample to prove that a statement is false. Teacher Preparation and Notes • This activity is designed to be used in a high school or middle school geometry classroom. • Before beginning this activity, students should be familiar with the terms inverse, converse, and contrapositive. • Students will discover the following concepts: — Parallel lines have slopes that are equal; perpendicular lines have slopes that are opposite recip- rocals (the product of the slopes is -1). — The Segment Addition Postulate states that AB + BC = AC if B is between A and C and the points are collinear. If the points are not collinear, then AB + BC > AC. • If desired, teachers can explore which of the statements in the activity are also biconditional statements (definitions that are always true). • This activity is designed to be student- centered with the teacher acting as a facilitator while students work cooperatively. Use the following pages as a framework as to how the activity will progress. www.ckl2.org 8 • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail ?id=258#. • To download the calculator files , go to http://www. education, ti. com/calculators/downloads/ US/Activities/Detail?id=8746 and select COLSEG.8xv and NOCOLSEG.8xv. Associated Materials • Student Worksheet: Conditional Statements http://www.ckl2.org/flexr/chapter/9687 • Cabri Jr. Application . C0LSEG.8xv and N0C0LSEG.8xv Problem 1 — Slopes of lines To begin, students should open a new Cabri Jr. file. Step 1: Students will need to construct two parallel lines. First, a line needs to be constructed using the Line tool and a point not on the line using the Point tool. Step 2: Using the Parallel tool, a line parallel to the existing line should be constructed through the point not on the original line. Step 3: Students will find the slope of both lines by using the Slope tool (Measure > Slope). Students can now use the Hand tool to drag the original line or the point and observe the results. What is true of the slopes of parallel lines? Students should record their observations on the worksheet part A. They will write the converse, inverse, and contrapositive of the statement and determine the truth of each. Students will next construct perpendicular lines in a new Cabri Jr. file. Step 4: Again construct a line and a point not on the line. Have students select the Perp. tool to construct a line perpendicular to the existing line through the point not on the original line. Step 5: Students will find the slope of both lines. They should drag the original line or the point and observe the results. Step 6: Have students select Calculate to find the product of the slopes. They should again drag the line to observe the results. 9 www.ckl2.org What is true of the slopes of perpendicular lines? Students should record their observations, write conditional statements, and determine the truth of the statements on the worksheet in part B. On a new Cabri Jr. file, students will construct two lines that have the same y-intercept. If the axes are not currently showing, they should select Hide/Show > Axes. Step 7: Students should start by placing a point on the y-axis using the Point > Point On tool. Step 8: Make sure that students construct two different lines with the same y-intercept (the point previously created on the y-axis) and find the slopes of both lines. Note: If desired, students can display the equations of both lines using the Coord. & Eq. tool instead of the slopes. Step 9: Direct students to drag the lines and the y-intercept point and observe the changes in the coordinates and slopes. Students will write conditional statements on the worksheet part C. They should sketch counterexamples for any false statements. The screenshot at right shows a counterexample for the converse statement "If two lines have different slopes, then they have the same y— intercept." Problem 2 — Collinear and noncollinear segments Step 1: Distribute the file COLSEG to students. Points A, B, and C are constructed on a line and the lengths AB, BC, and AC are displayed. Ht-Z.Z IK.-I.B Step 2: Students will drag the points with the Hand tool and observe the changes in the lengths. Be sure that students drag all three points, and change the order of the points (they should investigate situations where B is not between the other two points). Students should record their observations on the worksheet and write a conditional statement. Step 3: Distribute the file NOCOLSEG to students. Points A, B, and C are not collinear. The lengths AB, BC, and AC are displayed. Step 4: Students should now drag the points and observe the changes in the distances. If desired, students can use the Calculate tool to find the sum of AS + BC. www.ckl2.org 10 Students should record observations on the worksheet and write a conditional statement. Solutions Problem 1 A. If two lines are parallel, then the slopes of the lines are equal. Converse: If the slopes of the lines are equal, then the two lines are parallel. Inverse: If the two lines are not parallel, then the slopes of the lines are not equal. Contrapositive: If the slopes of the lines are not equal, then the two lines are not parallel. Each conditional statement is true. B. If two lines are perpendicular, then the slopes of the lines are equal to -1. Converse: If the slopes of two lines are equal to -1, then the lines are perpendicular. Inverse: If two lines are not perpendicular, then the slopes of the lines are not equal to -1. Contrapositive: If the slopes of two lines are not equal to -1, then the lines are not perpendicular. Each conditional statement is true. C. Converse: If two lines have different slopes, then the lines have the same v-intercept. Inverse: If two different lines do not have the same v-intercept, then the lines do not have different slopes. Contrapositive: If two lines do not have different slopes, then the different lines do not have the same v-intercept. Each conditional statement is false. Problem 2 A. Answers will vary for distances of AB, BC, and AC. The lengths AB and BC add up to equal AC when B is between A and C. If A, B, and C are collinear and B is between A and C, then AB to +BC = AC. B. Answers will vary for distances of AB, BC, and AC. AB + BC> AC If A, B, and C are not collinear, then the sum of the lengths of AB and BC is greater than AC. 11 www.ckl2.org Chapter 4 TE Parallel and Perpendiculuar Lines - TI 4.1 Parallel Lines cut by a Transversal This activity is intended to supplement Geometry, Chapter 3, Lesson 2. ID: 8681 Time required: 30 minutes Topic: Points, Lines & Planes • Construct two parallel lines and an intersecting transversal and measure the angles to discover that pairs of alternate angles (interior and exterior) are equal and same side angles (interior and exterior) are supplementary. Activity Overview In this activity, students will develop and strengthen their knowledge about the angles formed when parallel lines are cut by a transversal. The measurement tool helps them discover the relationships between and among the various angles in the figure. Students will draw parallel lines, draw a transversal, and explore angle relationships. Teacher Preparation • This activity is designed to be used in a high-school geometry classroom. It can also be used with advanced middle school students. • Students should already be familiar with parallel lines as well as reading and naming angles. • The screenshots on pages 1-3 demonstrate expected student results. Classroom Management • This activity is intended to be mainly teacher-led, with breaks for individual student work. Use the following pages to present the material to the class and encourage discussion. Students will follow along using their calculators. www.ckl2.org 12 • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail ?id=258#. Associated Materials • Student Worksheet: Parallel Lines cut by a Transversal http://www.ckl2.org/flexr/chapter/ 9688 • Cabri Jr. Application See the student worksheet on how to complete this activity. 4.2 Transversals This activity is intended to supplement Geometry, Chapter 3, Lesson 3. ID: 10989 Time required: 15 minutes Activity Overview In this activity, students will explore corresponding, alternate interior and same-side interior angles. This is an introductory activity where students will need to know how to grab points in Cabri Jr. Topic: Points, Lines & Planes • Corresponding angles are congruent • Alternate Interior angles are congruent • Same-Side Interior angles are supplementary Teacher Preparation and Notes • This activity was written to be explored with Cabri Jr. • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. • To download the calculator file, go to http: //www . education, ti . com/calculators/downloads/ US/Activities/Detail?id=10989 and select TRNSVRSL.8xv. Associated Materials • Student Worksheet: Transversals http://www.ckl2.org/flexr/chapter/9688, scroll down to the second activity. • Cabri Jr. Application . TRNSVRSL.8xv Exploring Parallel Lines cut by a Transversal Before beginning the activity, students should know the definition of corresponding, alternate interior and same-side interior angles. Questions 1, 2, and 3 ask students to name pairs of angles from the diagram. This should be done without the use of the calculator. 13 www.ckl2.org -oE-- Students will now run the Cabri Jr. App and open the file TRNSVRSL. To open a file, they should press Y = and select Open. By moving point G, students will discover the properties of two parallel lines cut by a transversal. To move a point, students need to move the cursor over the point (a square) and press ALPHA. For Questions 4, 5, and 6, students will move point G to four different places. They should record the angle measurements in the tables on the worksheet. Then, students should try to generalize their results in the Conjecture section. There are two application problems at the end of the worksheet for students to apply what they have learned in the activity. These problems can be done as homework. Solutions 1. Z4 and Z5 is another pair 2. Z4 and Z6 is another pair 3. Z4 and Z8 is another pair 4. a. Corresponding b. Sample measurements. Table 4.1: 1 st position 2 position 3 position 4 position mlABC mlHFB 109 109 84 84 56 56 37 37 c. Congruent 5. a. Same-Side Interior b. Sample measurements. Table 4.2: I st position 2 position 3 position 4 position mlABF mlHFB 150 30 136 44 112 68 75 105 c. Supplementary 6. a. Alternate Interior www.ckl2.org 14 b. Sample measurements. Table 4.3: 1 st position 2 position 3 position 4 position mlDBF mlBFH 43 43 60 60 108 108 125 125 c. Congruent Conjectures For parallel lines and a transversal... 7. if two angles are corresponding angles, then they are congruent. 8. if two angles are alternate interior angles, then they are congruent. 9. if two angles are same-side interior angles, then they are supplementary. Extra Problems 10. Zl, Z2, and Z3 are all equal to 55° 11. 108 = 7x - 4 and y 112 = 7x 16 = x 72 4.3 Perpendicular Slopes This activity is intended to supplement Geometry, Chapter 3, Lesson 4- ID: 8973 Time required: 45 minutes Topic: Linear Functions • Graph lines whose slopes are negative reciprocals and measure the angles to verify they are perpen- dicular. Activity Overview In this activity, students investigate the "negative reciprocal" relationship between the slopes of perpendicular lines. The final phase of the activity is appropriate for more advanced students as they are led through an algebraic proof of the relationship. Optional geometric activities (Problems 5 and 6) use the result to verify that (1) the radius of a circle and its tangent line are perpendicular and (2) a triangle inscribed in a circle with the diameter as one side is a right triangle. Teacher Preparation This activity is appropriate for students in Algebra 1. It is assumed that students have recently been introduced to the notion of slope and perhaps the fact that two lines are parallel if and only if they have the same slope. 15 www.ckl2.org Classroom Management • This activity is designed to have students explore individually and in pairs. However, an alternate approach would be to use the activity in a whole-class format. By using the computer software and the questions found on the student worksheet, you can lead an interactive class discussion on the slope of perpendicular lines. • This worksheet is intended to guide students through the main ideas of the activity. You may wish to have the class record their answers on a separate sheet of paper, or just use the questions posed to engage a class discussion. • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. • To download the calculator files , go to http: //www . education, ti. com/calculators/downloads/ US/Activities/Detail?id=8973 and select PERP1-PERP6 Associated Materials • Student Worksheet: Perpendicular Slopes http : //www . ckl2 . org/f lexr/chapter/9688, scroll down to the third activity. • Cabri Jr. Application . PERP1.8xv, PERP2.8xv, PERP3.8xv, PERP4.8xv, PERP5.8xv, and PERP6.8xv See the student worksheet on how to complete this activity. www.ckl2.org 16 Chapter 5 TE Congruent Triangles - TI 5.1 Interior and Exterior Angles of a Triangle This activity is intended to supplement Geometry, Chapter 4; Lesson 1. ID: 8775 Time required: 40 minutes Activity Overview In this activity, students will measure interior and exterior angles of a triangle and make conjectures about their relationships. Topic: Triangles & Congruence • Use inductive reasoning to conjecture a theorem about the total measures of a triangle's interior angles. • Prove that the sum of the measures of the interior angles of a triangle is 180°. • Prove that the sum of the measures of the exterior angles of a triangle is 360°. Teacher Preparation • This activity is designed to be used in a high school or middle school geometry classroom. • This activity is designed to be student- centered with the teacher acting as a facilitator while students work cooperatively. Use the following pages as a framework as to how the activity will progress. • If needed, review with students the definitions of the following angles before the activity: Interior angle, exterior angle, remote interior angle, adjacent interior angle. • Note: Measurements can display 0, 1, or 2 decimal digits. If digits are displayed, the value shown will round from the actual value. To change the number of digits displayed: 1. Move the cursor over the coordinate value so it is highlighted. 2. Press + to display additional decimal digits or - to hide digits. • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. 17 www.ckl2.org Associated Materials • Student Worksheet: Interior and Exterior Angles http://www.ckl2.org/flexr/chapter/9689. • Cabri Jr. Application Problem 1 — Interior angles of a triangle Students should open a new Cabri Jr. file. They will construct a triangle using the Triangle tool. A ^ i Select the Alph-Num tool to label the vertices A, B, and C as shown. Note: Press ENTER to start the label, then press ENTER again to end the label. Have students measure the three interior angles of the triangle using the Measure > Angle tool. Note: To measure an angle, press ENTER on three points, with the vertex of the angle being the second point selected. Students should records this data in the first row of the chart on the student worksheet. Students should drag a vertex of the triangle to change the angle measures. Have them try to create different types of triangles (acute, obtuse, right) and record two more sets of data in the chart. From here, they should make a conjecture about the three interior angles. ,-** ■"■ 55^ j^ i c Instruct students to use the Calculate tool to find the sum of the three interior angles of the triangle. Drag a vertex and observe the results. Ask: Do the results support your conjecture? Have students save the file as IntAngle. Problem 2 — One exterior angle of a triangle Students should continue using the previous file, but instruct them to save the file as ExtAngle using the Save As option. Have students construct a line through the two lower vertices of the triangle (A and C) using the Line tool. Note: To be certain that the line passes through a vertex, be sure that the vertex point is flashing before pressing ENTER. www.ckl2.org 18 Next, students should create a new point on the line to the right of the triangle using the Point > Point On tool. Label it D. A / \ /'H7» 7i-> ^ "k Direct students to measure the exterior angle IBCD with the Measure > Angle tool. Record this measure and the measures of the interior angles on the chart on the student worksheet. They can then drag a vertex of the triangle to change the angle measures, and add two more sets of data to the chart. Students should now make some observations about the exterior angle and its relationship to other angles in the chart. Encourage them to make calculations as needed to test their conjectures. If needed, suggest that they calculate the sums of pairs of angles in the chart. Again, students can drag a vertex and observe the results. Ask: Do the results support your conjectures? Before proceeding, have students save the file as ExtAngle. Problem 3 — Three exterior angles of a triangle Students should continue using the previous file, but instruct them to save the file as ExtAng3 using the Save As tool. Note: Press the ALPHA button to access the numeric character "3." Students will select the Line tool and construct a line through vertices A and B of the triangle as shown below. Next, students should measure one exterior angle at each vertex. Note: It is not necessary to create an additional point on the line before measuring the angle. y M 117* /A / \ _i33y nr» ?±> nag* _ DP D D — C [■ 7* Have them record the measures of the three exterior angles into the chart on the student worksheet. They can then drag a vertex of the triangle and record more data into the chart. Instruct students to make a conjecture about the three exterior angles, and have them calculate the sum of the three exterior angles. Tell them to drag a vertex and observe the results. Ask: Do the results support your conjectures? 19 www.ckl2.org Solutions 1. IB = 43°, IBCD = 143° 2. IBCA = 117°,Z5 = 32° 3. IBCA = 81°,int lA = 42°,ext lA = 138° 4. IBCA = 83°,ext Z/4 = 140°, lABC = 57°,ext Zfl = 123° 5.2 Congruent Triangles This activity is intended to supplement Geometry, Chapter 4> Lesson 4- ID: 8817 Time required: 45 minutes Topic: Triangles & Congruence • Investigate the SSS, SAS, and ASA sets of conditions for congruent triangles. Activity Overview In this activity, students will explore the results when a new triangle is created from an original triangle using the SSS, SAS, and ASA sets of conditions for congruence. In doing so, they will use the Cabri Jr. Compass tool to copy a segment and the Rotation tool to copy an angle. Teacher Preparation • This activity is designed to be used in a high school or middle school geometry classroom. • The Compass tool copies a given length as a radius of a circle to a new location. • The screenshots on pages 1-1 demonstrate expected student results. Classroom Management • This activity is designed to be student- centered with the teacher acting as a facilitator while students work cooperatively. Use the following pages as a framework as to how the activity will progress. It is recommended that you print out the following pages as a reference for students. • If desired, have students work in groups with each group focusing on one of the problems. • The student worksheet helps guide students through the activity and provides a place for students to record their answers and observations. • The student worksheet includes an optional extension problem to investigate two corresponding sides and the NON-included angle. • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. Associated Materials • Student Worksheet: Congruent Triangles http://www.ckl2.org/flexr/chapter/9689, scroll down to the second activity. • Cabri Jr. Application www.ckl2.org 20 Problem 1 — Three Corresponding Sides (SSS) Step 1: Students should open a new Cabri Jr. file. They will first construct a scalene triangle using the Triangle tool. Step 2: Select the Alph-Num tool to label the vertices A, B, and C as shown. / Step 3: Have students construct a free point on the screen using the Point tool. Label it point D with the Alph-Num tool. Step 4: Students will select the Compass tool to copy AB to point D. • Press ENTER on AB. A dashed circle will appear and follow the pointer. • Press ENTER on point D. The compass circle is anchored at center D. Have students construct a point on the compass circle and label it point E. f / ,' s~~ ~~\ p k ,. r '■■--_---' fc Step 5: Direct students to create DE with the segment tool. This segment is a copy of AB. Hide the compass circle with the Hide/Show > Object tool. Save this file as CongTri. This setup will be used again for Problems 2 and 3. Step 6: Drag point E, and observe that the location of DE can change but it will not change in length. You must drag either point A or point B to change the length of DE. Step 7: To copy the other two segments of AABC, repeat the use of the Compass tool. Students should copy AC to point D. Students should copy BC to point E. Find the intersection point of the two compass circles and hide the circles. Label the intersection point as F. Finally, construct segments DF and EF to complete the new triangle. 21 www.ckl2.org Step 8: Students will now investigate whether ADEF is congruent to AABC. Measure the sides and angles of both triangles to confirm that all corresponding parts are congruent. Use the Measure > D. & Length and Measure > Angle tools. Step 9: Drag vertices of the triangles and observe the results. Dragging vertices of AABC will cause both triangles to change in size and shape. Dragging vertices of ADEF will cause ADEF to change its location but its size and shape will remain the same as AABC. Notice that not all vertices of ADEF are available to be dragged, since they have been constructed to be dependent on AABC. If students wish to save their work, use the Save As tool and do not reuse the name CongTri. Problem 2 — Two Corresponding Sides and the Included Angle (SAS) Step 1: Students should open the Cabri Jr. file CongTri that they created in Problem 1. Recall that DE is a copy of AB. Step 2: Students will select the Rotation tool to copy lABC. • Press ENTER on point E. This is the center of rotation. • Press ENTER on DE. This is the object to be rotated. • Press ENTER three times on the vertices A, B, and C in that order to identify the angle of rotation. A new rotated segment appears. / /f J E k Step 3: Have students use the Line tool to construct a line from point E through the endpoint of the new segment. Use the Hide/Show > Object tool to hide the endpoint of the rotated segment. Step 4: Use the Compass tool to copy BC to point E. Create the intersection point of the compass circle and the line and label it point F . Hide the compass circle and the line. Finally, construct DF and EF to complete the new triangle. www.ckl2.org 22 I\ / ™ . '* 1 [.^"" iE 1 * Step 5: Students will now investigate whether ADEF is congruent to AABC. Measure the sides and angles of both triangles to confirm that all corresponding parts are congruent. Drag vertices of the triangles and observe the results. If students wish to save their work, use the Save As tool and do not reuse the name CongTri. Problem 3 — Two Corresponding Angles and the Included Side (ASA) Step 1: Students should open the Cabri Jr. file CongTri that they created in Problem 1. Recall that DE is a copy of AB. Step 2: Students will select the Rotation tool to copy lABC. • Press ENTER on point E. This is the center of rotation. . Press ENTER on ~DE. This is the object to be rotated. Press ENTER three times on the vertices A, B, and C in that order to identify the angle of rotation. A new rotated segment appears. Step 3: Students will select the Rotation tool to copy IBAC. • Press ENTER on point D. This is the center of rotation. • Press ENTER on DE. This is the object to be rotated. • Press ENTER three times on the vertices B, A, and C in that order to identify the angle of rotation. A new rotated segment appears. ?' : I /J, Step 4: Have students use the Line tool to construct lines over the new segments. Use the Hide/Show > Object tool to hide the endpoints of the rotated segments. 23 www.ckl2.org Step 5: Create the intersection point of the two lines and label it point F. Hide the lines. Finally, construct DF and EF to complete the new triangle. Step 6: Students will now investigate whether ADEF is congruent to AABC. Measure the sides and angles of both triangles to confirm that all corresponding parts are congruent. Drag vertices of the triangles and observe the results. If students wish to save their work, use the Save As tool and do not reuse the name CongTri. 5.3 Triangle Sides and Angles ID: 8795 Time required: 40 minutes Topic: Triangles &: Congruence • Classify triangles by angle measure. • Prove and apply the Isosceles Triangle Theorem. • Prove and apply the converse of the Isosceles Triangle Theorem. • Recognize the relationship between the side lengths and angle measures of a triangle. Activity Overview In this activity, students will explore side and angle relationships in a triangle. First, students will discover where the longest (and shortest) side is located relative to the largest (and smallest) angle. Then, students will explore the Isosceles Triangle Theorem and its converse. Finally, students will determine the number of acute, right, or obtuse angles that can exist in any one triangle. Teacher Preparation • This activity is designed to be used in a high school or middle school geometry classroom. • The screenshots on pages 1-3 demonstrate expected student results. Classroom Management • This activity is designed to be student- centered with the teacher acting as a facilitator while students work cooperatively. Use the following pages as a framework as to how the activity will progress. • The student worksheet helps guide students through the activity and provides a place for students to record their answers and observations. • Note: Measurements can display 0, 1, or 2 decimal digits. If digits are displayed, the value shown will round from the actual value. To change the number of digits displayed: www.ckl2.org 24 1. Move the cursor over the coordinate value so it is highlighted. 2. Press + to display additional decimal digits or - to hide digits. • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. Associated Materials • Student Worksheet: Triangle Sides and Angles http://www.ckl2.org/flexr/chapter/9689, scroll down to the third activity. • Cabri Jr. Application Problem 1 — Size and Location of Sides and Angles Step 1: Opening a new Cabri Jr. file, students should first construct a triangle using the Triangle tool. Step 2: Select the Alph-Num tool to label the vertices A, B, and C as shown. Note: Press ENTER to start entering the label, and then press ENTER again to end the label. A *£ h H C Step 3: Students should measure the three interior angles of the triangle using the Measure > Angle tool. To measure an angle, press ENTER on each of three points that define the angle, with the vertex of the angle as the second point chosen. Step 4: Direct students to measure the three side lengths using the Measure > D. & Length tool. Note: To measure a side of the triangle, you must click on the endpoints of the segment. If you click on the side itself, the tool will return the perimeter of the triangle. Step 5: Now students should drag a vertex of the triangle to change the angle measures and side lengths. Encourage them to make a conjecture about the sizes and locations of the angles and sides in a triangle. Students can answer the question on the worksheet based on their observations, concluding that the largest angle is opposite the longest side, and the smallest angle is opposite the shortest side. Step 6: Have students save this file as Triangle. Problem 2 —The Isosceles Triangle Theorem Step 1: Distribute the Cabri Jr. files Isostril and Isostri2 to student calculators Have students open the file Isostril. An isosceles triangle has been constructed and the congruent sides have side lengths displayed. Step 2: Students should measure all three angles using the Measure > Angle tool. Step 3: Direct students to drag a vertex of the triangle to explore what happens to the angle measures when two sides have equal lengths. 25 www.ckl2.org Students should conclude that a triangle with two congruent sides also has two congruent angles. This is known as the Isosceles Triangle Theorem. Step 4: The converse of the Isosceles Triangle Theorem may be explored in the file Isostri2. Step 5: Students should measure all three sides using the Measure > Length tool, then drag a vertex to explore. Step 6: On their worksheets, students should make the conclusion that a triangle with two congruent angles also has two congruent sides. Problem 3 — Types of Angles in a Triangle Step 1: Tell students to open the file Triangle that they saved from Problem 1. If desired, have them hide the side lengths using the Hide/Show > Object tool. Step 2: Students should drag a vertex and notice how many angles of each type can exist in a triangle. They should conclude that a triangle: • can have three acute angles • cannot have three right angles • cannot have three obtuse angles Challenge them to observe that a triangle can have, at most, one right or one obtuse angle and that a triangle cannot have both a right and obtuse angle. www.ckl2.org 26 Chapter 6 TE Relationships within Triangles - TI 6.1 Perpendicular Bisector This activity is intended to supplement Geometry, Chapter 5, Lesson 2. ID: 11198 Time required: 15 minutes Activity Overview In this activity, students will explore the perpendicular bisector theorem and discover that if a point is on the perpendicular bisector of a segment, then the point is equidistant from the endpoints. This is an introductory activity where students will need to know how to grab and move points, measure lengths, and construct the perpendicular bisector with Cabri Jr. Topic: Triangles &: Their Centers • Perpendicular Bisector Theorem Teacher Preparation and Notes • This activity was written to be explored with the Cabri Jr. on the TI-84- • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. • To download the calculator files , go to http: //www . education, ti. com/calculators/downloads/ US/Activities/Detail?id=11198 and select PERPBIS and POINTS. Associated Materials • Student Worksheet: Perpendicular Bisectors http://www.ckl2.org/flexr/chapter/9690 • Cabri Jr. Application . PERPBIS.8xv and POINTS.8xv 27 www.ckl2.org In this activity students will explore the properties of perpendicular bisectors. In the first part, students will measure the distance to a point on the bisector to identify congruent segments. In the second part, students will apply what they have learned about perpendicular bisectors to the context of map coordinates. Problem 1 — Exploring the Perpendicular Bisector Theorem Students will be exploring the distance from a point on the perpendicular bisector to the endpoints of a segment and discover that if a point is on the perpendicular bisector of a segment, then it is equidistant from the two endpoints of the segment. Students will measure the lengths of segments using the Distance & Length tool (press GRAPH and select Measure > D.&Length). To drag a point, students will move the cursor over the point, press ALPHA, move the point to the desired location, and then press ALPHA again to release the point. > Is \ k i y A rf- \ >, Problem 2 — An Application of the Perpendicular Bisector The- orem Students will apply what they learned from problem one and find the intersection of two perpendicular bisectors to find a point equidistant from three points. To draw a segment, students should use the Segment tool (press WINDOW and select Segment). To draw a perpendicular bisector, press ZOOM and select Perp. Bis.. I.K..VJ Cl/J.53q V C5*» S& Solutions Problem 1 — Exploring the Perpendicular Bisector Theorem • Answers will vary. The measures of AC and BC should be equal. • Sample answer: AC is congruent to BC. The distances are the same. • If a point is on the perpendicular bisector of a segment, then it is equidistant from the two endpoints of the segment. Problem 2 — An Application of the Perpendicular Bisector Theorem www.ckl2.org 28 (6,7) Fl 6.2 Hanging with the Incenter This activity is intended to supplement Geometry, Chapter 5, Lesson 3. ID: 11360 Time Required: 45 minutes Activity Overview In this activity, students will explore the angle bisector theorem and discover that if a point is on the angle bisector of a segment, then the point is equidistant from the sides. This is an introductory activity, but students will need to know how to grab and move points, measure lengths, and construct the perpendicular bisector with Cabri Jr. Topic: Triangles &: Their Centers • Angle Bisector Theorem • Incenter Teacher Preparation and Notes • This activity was written to be explored with the Cabri Jr. on the TI-84- • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. • To download the calculator file, go to http://www. education, ti . com/calculators/downloads/ US/ 'Activities/Detail? id=11360 and select ANGBIS. Associated Materials • Student Worksheet: Hanging with the Incenter http://www.ckl2.org/flexr/chapter/9690, scroll down to the second activity. • Cabri Jr. Application . ANGBIS.8xv Problem 1 — Exploring the Angle Bisector Theorem Students will be exploring the distance from a point on the angle bisector to the segment. They will discover that if a point is on the angle bisector of an angle, then it is equidistant from the two segments of the angle. Students will measure the lengths of segments using the Distance & Length tool (press GRAPH and select Measure > D.& Length). To drag a point, students will move the cursor over the point, press ALPHA, move the point to the desired location, and then press ALPHA again to release the point. 29 www.ckl2.org Problem 2 — Exploring the Incenter of a Triangle Students will need to open a new Cabri Jr. file by pressing Y =, selecting New, and answer no if asked to save. Students are to create an acute triangle and find the angle bisector of all three angles of the triangle. Students should realize that they are concurrent and answer Questions 4-8 on the accompanying worksheet. Students will need to find the distance from the incenter to the 3 sides of the triangle. Students may need to be reminded how to find the distance from a point to a line or segment. It is the perpendicular distance from the incenter to the side. Teachers should encourage the students to hide their angle bisectors. Be sure that all students understand that the perpendicular line created from the incenter to a side of the triangle is not the perpendicular bisector. Students should discover that all of these distances are equal. Problem 3 — Extension Problem 3 is an extension of this activity. Students will use the handheld and the angle bisectors to find the coordinates of fence posts on a plot of land. Note: This problem will be very hard for the average student. Students must first set up coordinates for the 2 fences at (-1, 2), (1, 0), (1, 3), and (5, 3). Next, students should create segments for the two fences. From here, there are several approaches to solving this problem. Using the incenter: One approach is to find the incenter of the triangle formed by the intersection of the two fences (over the pond) and a segment joining the two fences. Students should construct a segment from (1, 0) to (5, 3) connecting the two fences. This is to create two angles that they can use to find angle bisectors. The point of intersection of the two angle bisectors is the incenter of the triangle. This point is therefore equidistant from the two fences. However, we need two points to determine the line where the fence will be. Students will need to create a similar triangle that connects the two fences. To guarantee similarity, choose a segment that is parallel to the segment joining (1, 0) and (5, 3) using the Parallel tool (Press ZOOM, then scroll down to Parallel). Find the incenter of the second triangle and you can create the line through those two incenters. One point on the line should be at (5.24, 0) or (5.2, 0) and the other can be any other point on the line connecting the two incenters. The y-intercept is another convenient point to find the coordinates of (0, 2.2). Using only an angle bisector: Another approach is to find the point of intersection of the fences (if they were continued over the pond) . Students can construct lines for the Fence 1 and Fence 2 and plot the intersection point of the lines. Then, students can use the Angle Bisector tool to find the angle bisector of this angle. This ray will be equidistant from Fence 1 and Fence 2. Students can plot a point on this ray and display and record the coordinates of this point. Then, move the point and record the new coordinates. www.ckl2.org 30 i '.l.v/sMS.ifjs.v K \ Solutions 1. Sample answers Table 6.1: Position 1 st position 2 nd position 3 rd position 4 th position DF DE 2.25473 2.25473 2.72284 2.72284 1.85757 1,85757 1.12521 1.12521 2. They are equal. 3. Equidistant 4. They are concurrent. 5. Not possible 6. Not possible 7. Obtuse, Acute, and Right 8. They are all equal. 9. Sample answer: (5.2, 0) and (0, 2.2) 6.3 Balancing Point This activity is intended to supplement Geometry, Chapter 5, Lesson 4- ID: 11403 Time Required: 45 minutes Activity Overview In this activity, students will explore the median and the centroid of a triangle. Students will discover that the medians of a triangle are concurrent. The point of concurrency is the centroid. Students should discover that a triangle's center of mass and centroid are the same. Topic: Triangles &: Their Centers • medians • centroid Teacher Preparation and Notes 31 www.ckl2.org This activity was written to be explored with the TI-84 using the Cabri Jr. application. This is an introductory activity for which students will need to know how to construct triangles, grab and move points, measure lengths, and construct segments. To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. To download the calculator files, go to http://www. education, ti. com/calculators/downloads/ US/Activities/Detail?id=11403 and select CENTROID, MEDIAL, and MIDSEG. Associated Materials Student Worksheet: Balancing Point http://www.ckl2.org/flexr/chapter/9690, scroll down to the third activity. Cabri Jr. Application CENTR0ID.8xv, MEDIAL.8xv, MIDSEG.8xv Problem 1 — Exploring the Centroid of a Triangle Students are to cut out the triangle on their worksheet and attempt to balance it with their pencil. Students will mark the balancing point on their triangle. Students will then use their handhelds to find the point where the triangle will be balanced. They will be asked to compare their findings to see if their point was close to the point given by the calculator. Explain to the students that the balancing point for an object is called the center of mass. Problem 2 — Exploring the Medians of a Triangle Students will need to define the terms median of a triangle and centroid from either their textbook or another source. Students will then create the three medians of the triangle given in Centroid. 8xv. Students will need to find the midpoint of the three sides of a triangle. Students can access the Midpoint tool by selecting ZOOM > Midpoint. Then, students can click on a side of the triangle to place the midpoint. Next, students should construct the segment connecting point A and the opposite midpoint. Students can find the Segment tool by selecting WINDOW > Segment. They should repeat this for the other two medians. Students are asked several questions on their accompanying worksheet. Students should be able to find the centroid of a triangle and understand that this point is the center of mass. I & a ™ let i? f ^ — www.ckl2.org 32 Problem 3 — Extending the Centroid Students will extend the concept of the centroid in Problem 3. First, define the medial triangle to be the triangle formed by connecting the midpoints of the sides of a triangle. In Medial. 8 xv, students will see a triangle and its centroid. Student should construct the medial triangle and the centroid of the medial triangle. Students will notice that the centroid of the original triangle and the centroid of the medial triangle are the same. Problem 4 — Extending the Median In Problem 4, students will extend the concept of the median. Tell students that the midsegment is a line segment joining the midpoints of two sides of a triangle. In Midseg.8xv, students will see AABC with midpoints D, E, and F of sides AC, AB, and BC, respectively. Students should then construct the median that extends from point A to point F. Next, they should construct the intersection point, G, of the median and the midsegment. Students should find the lengths of AG, FG ,DG, and EG. Students can find the Length tool by selecting GRAPH > Measure > D. & Length. Students will notice that the median and midsegment bisect each other. Solutions 1. Student solutions will vary. Students' balancing points should be close to (2, 0). 2. The median of a triangle is the segment joining the vertices of a triangle to the midpoint of the opposite side. 3. They are concurrent. 4. (2, 0) 5. They are close. 6. (=^, =*£=*) = (2.0) 7. Centroid is the point of concurrency of the medians. 8. They are the same. 9. They bisect each other. 6.4 Hey Ortho! What's your Altitude? This activity is intended to supplement Geometry, Chapter 5, Lesson 4- ID: 11483 Time Required: 45 minutes Activity Overview In this activity, students will explore the altitudes of a triangle. Students will discover that the altitude can be inside, outside, or a side of the triangle. Students will discover that the altitudes are concurrent. The point of concurrency is the orthocenter. Students should discover the relationship between the type of 33 www.ckl2.org triangle and the location of the point of concurrency. Students will discover properties of the orthocenter in equilateral triangles. Topic: Triangles &: Their Centers • Altitudes of a Triangle • Orthocenter Teacher Preparation and Notes • This activity was written to be explored with the Cabri Jr. on the TI-84- • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. • To download the calculator files , go to http://www. education, ti. com/calculators/downloads/ US/Activities/Detail?id=11483 and select ACUTE, EQUILATE, MEDIAL2, OBTUSE and TRIANGLE. Associated Materials • Student Worksheet: Hey Ortho! Whats your Altitude? http://www.ckl2.org/flexr/chapter/ 9690, scroll down to the fourth activity. • Cabri Jr. Application . ACUTE.8xv, OBTUSE.8xv, TRIANGLE.8xv, EQUILATE.8xv, and MEDIAL2.8xv Problem 1 — Exploring the Altitude of a Triangle Students should define the altitude of a triangle using their textbooks or other source on their worksheet. In files ACUTE and OBTUSE, students are to create the altitude of an acute triangle, obtuse triangle, and right triangle, respectively. Students will need to construct the perpendicular line to a line through a point (press ZOOM and select Perp.). To show the altitude of the triangle, students will need to find the intersection point of the perpendicular line and the line that extends from the base of the triangle. Students will then need to create a segment from the opposite vertex to the intersection point. Problem 2 — Exploring the Orthocenter In file TRIANGLE, students are given AABC. They should construct the altitude of each vertex of the tri- angle. Students should realize that they are concurrent. Explain to students that the point of concurrency is called the orthocenter of the triangle. Students should discover a few facts about the orthocenter. If a triangle is an acute triangle, then the orthocenter is inside of the triangle. If a triangle is a right triangle, then the orthocenter is on a side of the triangle. If a triangle is an obtuse triangle, then the orthocenter is outside of the triangle. / .-■■* T-vti.. -■ s i, --s _-- T www.ckl2.org 34 Problem 3 — Exploring the Altitude of an Equilateral Triangle Students will need to find the distance from point P to the 3 sides of the triangle and the altitude BD in the file EQUILATE. The Distance & Length tool is found in the GRAPH/F5 menu (press GRAPH and select Measure > D.&Tength). Once students make their calculation, they can move it to the right of the appropriate label (EP, for instance). Students will then need to calculate the sum of EP + FP + GP using the Calculate tool (press GRAPH and select Calculate). Students will need to move their cursor to EP until it is underlined, then press ENTER, then press +, then move their cursor to FP until it is underlined, then press ENTER. A measurement that you can place anywhere on the screen is given. Place the measurement by pressing ENTER. Repeat this process for EP + FP that was just created and GP to get the sum EP + FP + GP. Students should discover that the sum of the distances from all three sides to point P and the length of the altitude BD are equal. Make sure that students see that this only works for an equilateral triangle. Problem 4 — Exploring the Orthocenter of a Medial Triangle In the file MEDIAL2, students are given a triangle, its medial triangle, and the orthocenter of the medial triangle. Students are to discover which point of the original AABC is the orthocenter of the medial triangle. Choices should include the centroid, circumcenter, incenter, and orthocenter. Students should discover that the orthocenter of the medial triangle is the circumcenter of the original triangle. 1 ,,,* "'■-.. if-" '<£&<*> Solutions 1. An altitude of a triangle is a perpendicular segment from a vertex to the line containing the side opposite that vertex. 2. A C 3. a. inside b. outside c. a side of 4. They are concurrent 5. Right Triangle 6. Acute Triangle 35 www.ckl2.org 7. Obtuse Triangle Table 6.2: Position 1 st position 2 nd position 3 rd position A th position BD EP + FP + GP 4.9 4.9 5.4 5.4 5.4 5.4 5.4 5.4 9. They are equal 10. equal to the length of the altitude 11. circumcenter www.ckl2.org 36 Chapter 7 TE Quadrilaterals - TI 7.1 Properties of Parallelograms This activity is intended to supplement Geometry, Chapter 6, Lesson 2. ID: 11932 Time Required: 15 minutes Activity Overview Students will explore the various properties of parallelograms. As an extension, students can also explore necessary and sufficient conditions that guarantee that a quadrilateral is a parallelogram. Topic: Quadrilaterals & General Polygons • Inductive Reasoning • Parallelograms Teacher Preparation and Notes • This activity was written to be explored with the Cabri Jr. app on the TI-84- • Before beginning this activity, make sure that all students have the Cabri Jr. application, and the Cabri Jr. files PAR1.8xv, PAR2.8xv, PAR3.8xv loaded on their TI-84 calculators. • In Cabri Jr., to grab a point hover the cursor over the point and press ALPHA. To release press ALPHA or ENTER. To move a point after grabbing it, use the arrow keys. • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. • To download the calculator files, go to http: //www . education, ti. com/calculators/downloads/ US/Activities/Detail?id=11932 and select PAR, PAR2, and PAR3. Associated Materials • Student Worksheet: Properties of Parallelograms http://www.ckl2.org/flexr/chapter/9691 • Cabri Jr. Application . PAR.8xv, PAR2.8xv, and PAR3.8xv 37 www.ckl2.org Problem 1 — Properties of Parallelograms Students will begin this activity by looking at properties of parallelograms. They will discover that opposite sides are congruent, opposite angles are congruent, and that consecutive angles are supplementary. As an extension, students can prove each of these using parallel lines and transversals. Students will need to know the properties of alternate interior angles, same-side interior angles, and corresponding angles. Problem 2 — Diagonals of Parallelograms In Problem 2, students are asked to investigate the diagonals of a parallelogram. Students should discover that the diagonals of a parallelogram bisect each other. This particular wording may be hard for students discover independently. Problem 3 — Extension: Proving Parallelograms In this problem, students can explore various properties and see if they guarantee that a quadrilateral is a parallelogram. Students should know that the following prove that a quadrilateral is a parallelogram: 1. both pairs of opposite sides are congruent 2. both pairs of opposite angles are congruent 3. both pairs of opposite sides are parallel 4. one pair of opposite sides is both parallel and congruent 5. the diagonals bisect each other Problem 4 — Extension: Extending the Properties For this problem, students will create any quadrilateral and name it GEAR. Next, students will find the midpoint of each side and connect the midpoints to form a quadrilateral. Students will use the properties of a parallelogram to see that a parallelogram is always created. Solutions 1. A quadrilateral with both pairs of opposite sides parallel. 2. Sample answers: Table 7.1: Position QU UA AD DQ 1 4.50 2.44 4.50 2.44 2 4.50 2.94 4.50 2.94 3 4.90 2.94 4.90 2.94 4 5.50 2.94 5.50 2.94 3. The opposite sides are congruent. www.ckl2.org 38 4. Sample answers: Table 7.2: Position LQ L13 LA ID 1 100 80 100 80 2 106 74 106 74 3 124 56 124 56 4 77 103 77 103 5. The consecutive angles are supplementary. 6. The opposite angles are congruent. 7. Sample answers: Table 7.3: Position QR AR DR RU 1 2.84 2.84 3.29 3.29 2 2.94 2.94 3.45 3.45 3 3.06 3.06 3.58 3.58 4 3.19 3.19 3.71 3.71 8. The diagonals bisect each other. 9. Yes 10. No 11. Yes 12. Parallelogram 13. Sample: Both pairs of opposite sides are parallel. 7.2 Properties of Trapezoids and Isosceles Trape- zoids This activity is intended to supplement Geometry, Chapter 6, Lesson 5. ID: 9084 Time required: 25 minutes Topic: Quadrilaterals & General Polygons • Prove and apply theorems about the properties of rhombuses, kites and trapezoids. Activity Overview A trapezoid is a quadrilateral where one pair of sides is parallel while the other two sides are not. In an isosceles trapezoid the non-parallel sides are congruent. In this activity we will attempt to create an 39 www.ckl2.org isosceles trapezoid from an ordinary trapezoid, then approach the problem in a different manner and finally, examine the properties of trapezoids. Teacher Preparation This activity is designed to be used in a middle-school or high-school geometry classroom. • For this activity, students should know the definitions of a trapezoid and isosceles trapezoid. If they do not already know these terms, you can define them as they appear in the lesson, but allow extra time to do so. • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. Associated Materials • Student Worksheet: Properties of Trapezoids and Isosceles Trapezoids http : //www . ckl2 . org/f lexr/ chapter/9691, scroll down to the second activity. • Cabri Jr. Application Classroom Management • This activity is intended to be mainly teacher-led, with breaks for individual student work. Use the following pages to present the material to the class and encourage discussion. Students will follow along using their graphing calculators. • See the student worksheet on how to complete this activity. 7.3 Properties of Rhombi, Kites and Trapezoids ID: 12093 Time Required: 45 minutes Activity Overview In this activity, students will discover properties of the diagonals of rhombi and kites and properties of angles in rhombi, kites, and trapezoids. Topic: Quadrilaterals & General Polygons • Rhombi • Kites • Trapezoids Teacher Preparation and Notes • This activity was written to be explored with the Cabri Jr. app on the TI-84- • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail?id=258#. • To download the calculator files, go to http: //www . education, ti. com/ 'calculators /downloads/ US/Activities/Detail?id=12093 and select READ1, READ2, READ3, KING1, KING2, KING 3, and TRAP. www.ckl2.org 40 Associated Materials • Student Worksheet: Properties of Rhombi, Kites and Trapezoids http://www.ckl2.org/flexr/ chapter/9691, scroll down to the third activity. • Cabri Jr. Application . READ1.8xv, READ2.8xv, READ3.8xv . KING1.8xv, KING2.8xv, KING3.8xv . TRAP.8xv Problem 1 — Properties of Rhombi Students will begin this activity by looking at angle properties of rhombi. They are asked several questions about the angles and diagonals of a rhombus. An extension of this exercise would be to prove each of these using parallel lines and transversals. Students will need to know the properties of alternate interior angles, same-side interior angles, and corresponding angles. Problem 2 — Properties of Kites In Problem 2, students will be asked to begin looking at angle properties of kites. They are asked several questions about the angles and diagonals of kites. Problem 3 — Properties of Trapezoids In this problem, they will explore angle properties of trapezoids. Students are given trapezoid TRAP and the measure of angles T, R, A, and P. Students will move point R to four different positions and collect the measures of angles T, R, A, and P onto their accompanying worksheet. An extension of this exercise would be to prove leg angles are supplementary using parallel lines and transversals. Students will need to know the properties of alternate interior angles, same-side interior angles, and corresponding angles. Solutions 1. Sample answers: Table 7.4: Position LR LE lA LD 1 2 3 4 76 51 95 64 104 129 85 116 76 51 95 64 104 128 85 116 2. supplementary 3. congruent 41 www.ckl2.org 4. right angles 5. bisect 6. Sample answers: Table 7.5: Position IK Ll IN IG 1 2 3 4 69 71 61 37 130 116 123 142 32 58 53 38 130 116 123 142 7. One pair of opposite angles is congruent and one pair of opposite angles is not congruent. 8. right angles 9. One pair of opposite angles is bisected and one set of opposite angle is not. The pair that is not congruent is the one getting bisected. 10. Sample answers: Table 7.6: Position T 1 108 2 126 3 39 4 57 R 72 54 141 123 54 45 24 37 126 135 156 143 11. Leg angles are supplementary. www.ckl2.org 42 Chapter 8 TE Similarity - TI 8.1 Constructing Similar Triangles This activity is intended to supplement Geometry, Chapter 7, Lesson 4- ID: 8179 Time required: 30 minutes Activity Overview In this activity, students will investigate three different methods of constructing similar triangles. They will use the Dilation tool with the dilation point inside and outside of the triangle to investigate different relationships. Also, students will use the Parallel tool to construct two similar triangles from one triangle. Topic: Ratio, Proportion & Similarity • Use inductive reasoning to classify each set of conditions as necessary and/or sufficient for similarity: a) the side lengths of one triangle are equal to the corresponding side lengths of another triangle. b) two triangles of one triangle are congruent to two angles of another triangle. Teacher Preparation • This activity is designed to be used in a high school geometry classroom. • Before starting this activity, students should be familiar with the term dilation. • This activity is intended to be mainly teacher-led, with breaks for individual student work. Use the following pages to present the material to the class and encourage discussion. Students will follow along using their calculators. • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. Associated Materials • Student Worksheet: Constructing Similar Triangles http://www.ckl2.org/flexr/chapter/9692 • Cabri Jr. Application 43 www.ckl2.org Students need to press APPS and select CabriJr to start the application. When they open a new document, they need to make sure that the axes are hidden. If the axes are displayed, press GRAPH > Hide/Show > Axes. Explain to students that similar triangles are those that have the same shape but not necessarily the same size. Congruent triangles are a special type of similar triangle where corresponding sides are congruent. In similar triangles, corresponding angles are congruent but corresponding sides are proportional. In this activity, students will look at three methods of constructing similar triangles and will test these properties using dilations or stretches. In order to examine all of the sides and angles, students should work in groups of three. Have one student (Student A on the worksheet) in each group construct the first triangle and save it as "SIMTRI." This is saved in the TI-84 Plus family as an APPVAR. Student A needs to transfer the APPVAR to the other two students in the group. Problem 1 — Similar Triangles using Dilation Student A will use the Triangle tool to construct a triangle of any shape or size. Then they need to use the Alph-Num tool to label the vertices P, Q, and R. Student A should measure angle P and side PQ, Student B should measure angle Q and side QR, Student C should measure angle R and side PR. Note: To increase the number of digits for the length of the side, hover the cursor over the measurement and press +. With the Point tool, students on their own calculators will construct a point C in the center of the triangle. They will then use the Alph-Num tool to place the number 2 at the top of the screen. Explain to students that the point C will be the center of the dilation and the number 2 will be the scale factor. -i-il F-Q E.iH A, With the Dilation tool selected, students need to move the cursor to point C, the center of the dilation, and press ENTER. They should notice that the shape of the cursor changes, which is to indicate that everything will be stretched away from this point. Then students move the cursor to the perimeter of the triangle and press ENTER. Finally, move to the scale factor, 2, and press ENTER. Students will see that a new, larger triangle will appear outside of APQR. Have a prelimenary discussion with students about whether they think the new triangle is similar to APQR and how they can confrim their hypothesis. www.ckl2.org 44 A\ Instruct students to label this triangle as XYZ so that X corresponds to P, Y to Q and Z to R. Each person in the group should select and measure their appropriate angle and side in the new triangle. Students should answer Questions 1 and 2 on the worksheet comparing the two angles and two sides. Explain to students that XY = 2 PQ indicates that the sides have the ratio 2:1. If all three sides display the same result, then the sides are said to be proportional. To answer Questions 3 and 4 on the worksheet, students will observe the changes in the triangles as they drag a point in the original triangle and the point of dilation. Problem 2 — Different Scale Factors Students will continue using the same file. To change the scale factor, students must select the Alph-Num tool, move the cursor to the scale factor of 2 and press ENTER. Then they should delete 2 by pressing DEL, press ALPHA to change the character to a number, and enter 3. Students can now answer Question 5 on the worksheet. Now students will investigate another scale factor by changing it from 3 to 0.5. They should then answer Questions 6 and 7 on the worksheet, summarizing their observations of the triangles and their measure- ments. For the last investigation in this problem, students will look at the effect of a dilation when the center point C is outside of the pre-image triangle and a negative scale factor is used. Students will need to move APQR by moving the cursor to one side and when all sides of the triangle flash, press ALPHA and then use the arrow keys. Problem 3 — Similar Triangles with a Parallel Line Finally, students will look at a completely different method of constructing similar triangles. All students will need to open a new file. It is the teacher's decision to have them save the file. Student A should construct a triangle PQR and transfer it to the others in their group. They will all measure their same side and angle as before. ,.' ■■■, P fiH* PQ T-.72 _-^ F: „/- All students need to construct a point on PQ using the Point on tool and label it S using the Alph-Num tool. Using this point, they construct a line that is parallel to QR by selecting the Parallel tool, and then choosing the point and side. 45 www.ckl2.org With the Intersection tool, students need to find the intersection of side PR and the parallel line, and then use the Alph-Num tool to label it as T. The Hide/Show > Object tool enables students to hide the parallel line, and then construct segment S T with the Segment tool. If students can prove that all of the angles are congruent, they have completed their first step in proving that the two triangles are similar. With the Measure > D. & Length tool, students need to measure segments of the corresponding sides. In the screen to the right all three pairs of sides have been measured. To complete the proof, students will need to confirm that all of the sides are proportional. Using the Calcu- late tool, they can select a side measurement, press the operation key (-r) and then select the corresponding side measurement. If all three ratios are equivalent, then the sides are proportional and the two triangles are similar. 8.2 Side-Splitter Theorem This activity is intended to supplement Geometry, Chapter 7, Lesson 5. ID: 12318 Time Required: 15 minutes Activity Overview In this activity, students will explore the side-splitter theorem. Topic: Ratio, Proportion &; Similarity • Side-Splitter Theorem Teacher Preparation and Notes • This activity was written to be explored with the Cabri Jr. app on the TI-84- • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. • To download the calculator files , go to http: //www . education, ti. com/calculators/downloads/ US/ 'Activities /Detail? , id=12318 and select SIDESP1, SIDESP2, and SIDESP3. Associated Materials • Student Worksheet: Side-Splitter Theorem http : //www . ckl2 . org/f lexr/chapter/9692, scroll down to the second activity. • Cabri Jr. Application . SIDESP1.8xv, SIDESP2.8xv, and SIDESP3.8xv Problem 1 — Side-Splitter Theorem Students will begin this activity by looking at the side-splitter theorem. Students are given a triangle with a segment parallel to one side. They will discover that if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. www.ckl2.org 46 Students will be asked to collect data by moving point A and point D. Students are asked several questions about the relationships in the triangle. .PA AD H.N / DC i V ~T x HS H.S Sft i.h Problem 2 — Application of the Side-Splitter Theorem In Problem 2, students will be asked to apply the side-splitter theorem to several homework problems. Problem 3 — Extension of the Side-Splitter Theorem In Problem 3, students will discover the corollary to the Side-Splitter Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. Students are asked several questions about the corollary to the side-splitter theorem. Solutions 1. Sample answers: Table 8.1: Position AD DC AS SR AD DC AS SR 1 4.1 1.8 4.6 2.0 2.27 2.30 2 3 2.8 3.4 3.2 1.07 1.06 3 2.6 2.4 3.2 3.0 1.08 1.07 4 3.3 3.1 2.5 2.3 1.06 1.08 2. The ratios of the side lengths are equal. o ad _ AS °- DC ~ SR 4. The ratio remains the same. 5. The ratio changes when moving point D. 6. When you move the parallel line, you are changing the proportion between the upper and lower segments. When you move the point, the segments may get longer or shorter, but the proportion stays the same. 7. 12 8. 16.8 9. Sample answers: 47 www.ckl2.org Table 8.2: Position RN NO EA AS RN NO A5 1 2.1 1.5 2.4 1.8 1.4 1.4 2 2.1 1.3 2.7 1.7 1.6 1.6 3 1.9 1.5 2.4 2.0 1.2 1.2 4 2.7 0.7 3.5 0.9 4 4 10. The ratios are equal. 11. ^ and fr are congruent 8.3 Perspective Drawing TTiz's activity can be used to supplement the "Know What?" from Chapter 7, Lesson 6. 1 The student worksheets for this activity are at the end of Chapter 12 because this activity can also be used at the end of the year. The link is below. ID: 10033 Time required: 60 minutes Activity Overview In this activity, students draw figures in one- and two-point perspective and compare and contrast the two types of drawings. They then create an isometric drawing and compare it to their drawings in perspective. Topic: 3— Dimensional Geometry • Construct 3- dimensional prisms and pyramids. • Record the number of faces, edges, and vertices of prisms. Teacher Preparation and Notes • Perspective drawings can be taught at any time in a geometry curriculum, but are most appropriate after lessons on parallel and perpendicular lines, three-dimensional figures, and symmetry. • Throughout the activity, students use many drawing and construction tools, such as the Segment, Parallel, and Perpendicular tools. In this document, the first use of a tool is by name and accom- panied by its location within the menu structure. For subsequent uses, the tool may be mentioned by name or its function, and the menu location is omitted. • This activity is designed to have students explore individually or in pairs. • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail?id=258#. • To download the calculator files, go to http: //www . education, ti. com/calculators/downloads/ US/Activities/Detail?id=10033 and select RECPRISM, TRIPRISM, TWOPERSP, ISO- DRAW. Associated Materials www.ckl2.org 48 • Student Worksheet: Perspective Drawing http://www.ckl2.org/flexr/chapter/9697, scroll down to the third activity. • Cabri Jr. Application . RECPRISM.8xv, TRIPRISM.8xv, TW0PERSP.8xv, and IS0DRAW.8xv Problem 1 — One-point perspective Students will open the Cabri Jr. file RECPRISM and find a vaishing point A along the horizon and a rectangle BCDE. They are to use the Segment tool from the F2 menu to draw AB and AC. These two segments (and any other segment that joins a vertex of the prism to the vanishing point) are hereafter referred to as vanishing segments. Next, they will create FG such that F is between A and B, G is between A and C, and FG X BC (Refer to the diagram at the right). To do this, they will need to do the following: • Place a point F on AB. (F2 > Point > Point On) Construct a line through F parallel to BC. (F3 > Parallel) • Plot point G at the intersection of the parallel line and AC. (F2 > Point > Intersection) • Hide the line. (F5 > Hide/Show > Object) • Draw FG. Students can now hide the vanishing segments, AB and AC, and draw BF and CG. The front and top faces of the rectangular prism have been constructed. Ask students if they are satisfied that this drawing completely represents a rectangular prism. Have them drag point A (press APLHA to grab the point) to either the far left or far right and ask students what is wrong with the drawing. (The rectangular prism does not have a face on either side.) To complete the prism, students next need to construct AD and AE. Then, using the Perpendicular tool (F3 > Perp.), construct two lines perpendicular to FG — one through F and the other through G. The vertical "back" edges GH and FJ may then be created by plotting the intersection points, hiding the lines, and constructing the segments as needed. To show all six edges, students should draw a segment connecting H and /. Once all of the vanishing segments are hidden, students will find they need to also construct segments DH and EJ. H -Q- & n-'"'"Ji J yi H Depending on the topics you have recently covered in class, you can now have students identify parallel, intersecting, and skew lines; count faces, vertices, and edges; or discuss the similarity of rectangles BCDE and FGHJ. 49 www.ckl2.org When students have finished exploring their prisms, they may wish to hide or dash the edges that would not be visible if the figure were not transparent. The appearance of the edges may be altered using the Display tool (F5 > Display). Students are to open TRIPRISM and create a triangular prism in one-point perspective on their own. Be sure to circulate around the room and assist students as needed. Allow students a few minutes to explore the figure by dragging point A along the horizon or dragging an edge. Vertices B, C, and D can also be dragged on this figure. Problem 2 — Two-point perspective The Cabri Jr. file TWOPERSP contains two vanishing points, A and B, signifying this rectangular prism will be drawn in two-point perspective. The file also contains a vertical line segment, CD, which will be the front edge of the prism. CD is constructed so that dragging point C changes the segment's length and dragging point D translates the segment. Next, students will then use the Segment tool to draw the vanishing segments AC, AD, BC, and BD. Then they should use the Point On, Parallel, Intersection, Hide/Show, and Segment tools (as they did in Problem 1) to construct the vertical edges, EF and GH, of the left and right faces. The top and bottom edges of those faces, CE, CG, DF, and DH may be drawn as well. Students should now hide the four vanishing segments, and then draw four more: from E and F to B and from G and H to A. The intersection of the upper two vanishing segments and that of the lower two vanishing segments should be plotted and a segment drawn between them to form the last vertical edge, JK. I ~w The vanishing segments may then be hidden so that students can draw the remaining edges of the prism. As before, allow students some time to drag edges, vertices, and vanishing points, observing how the prism changes as they do so. (Dragging CD is very interesting!) Again, students can either hide or dash any "hidden edges." Problem 3 — An isometric drawing Using ISODRAW, students will construct an isometric drawing of a rectangular prism. The three seg- ments shown represent the three "front" edges of the prism, and each of the angles the segments form measures 120°. Students should first use the Parallel, Intersection, Hide/Show, and Segment tools to construct AE, CF, DE, and DF, as shown in the diagram to the right. They may then use the same sequence of tools once more to construct the remaining edges: AG, CG, EH, ~FH, and GH. www.ckl2.org 50 r After completing the isometric drawing, students should compare it to the perspective drawings from Problems 1 and 2. As a similarity, students may identify that right angles "in real-life" do not actually measure 90° in the drawings. As a difference, all parallel lines "in real-life" are parallel in an isometric drawing, whereas in perspective drawings, some parallel lines actually "meet" at the vanishing point. 51 www.ckl2.org Chapter 9 TE Right Triangle Trigonometry - TI 9.1 The Pythagorean Theorem This activity is intended to supplement Geometry, Chapter 8, Lesson 1. ID: 9532 Time required: 60 minutes Activity Overview In this activity, students will use the Cabri Jr. application to construct figures that prove the Pythagorean Theorem in two different ways. Topic: Right Triangles & Trigonometric Ratios • Construct and measure the side lengths of several right triangles and conjecture a relationship between the areas of the squares drawn on each side. • Prove and apply the Pythagorean Theorem. Teacher Preparation • This activity is designed to be used in a high school or middle school geometry classroom. • This activity is designed to be student- centered with the teacher acting as a facilitator while students work cooperatively. Use the following pages as a framework as to how the activity will progress. Feel free to print out the following pages for your students. • The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs. This can be expressed as c 2 = a 2 + b 2 where c is the length of the hypotenuse. • Depending on student skill level, you may wish to download the constructed figures to student cal- culators. If the files are downloaded, skip the construction steps for each problem and begin each at Step 10. • Note: Measurements can display 0, 1, or 2 decimal digits. If digits are displayed, the value shown will round from the actual value. To change the number of digits displayed: 1. Move the cursor over the value so it is highlighted. www.ckl2.org 52 2. Press + to display additional decimal digits or - to hide digits. • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail?id=258#. • To download the calculator files, go to http://www. education, ti. com/calculators/downloads/ US/Activities/Detail?id=9532 and select PYTHAG1 and PYTHAG2. Associated Materials • Student Worksheet: The Pythagorean Theorem http://www.ckl2.org/flexr/chapter/9693 • Cabri Jr. Application . PYTHAG1.8xv and PYTHAG2.8xv Problem 1 — Squares on Sides Proof The Pythagorean Theorem states that, the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the legs. In this activity, you will construct a right triangle and verify the Pythagorean Theorem by constructing squares on each side and comparing the sum of the area of the two smaller squares to the area of the square of the third side. Step 1: Open a new Cabri Jr. file. Construct a segment using the Segment tool. Select the Alph-Num tool to label the endpoints B and C as shown. Step 2: Construct a line through C that is perpendicular to BC using the Perp. tool. Step 3: Construct a point on the perpendicular line and label it A. Hide the perpendicular line with the Hide/Show tool and construct line segments AC and AB. For the time being, keep the sides of the triangle fairly small so that squares can be constructed on the sides. Step 4: In the lower left corner, use the Alph-Num tool to place the number 90 on the screen. This will be the angle of rotation. Note: Press ALPHA to access numerical characters. A small "1" will appear in the tool icon in the upper left corner of the screen. Step 5: Use the Rotation tool to rotate point C about point B through an angle of 90°. • Press ENTER on point B as the center of rotation. • Press ENTER on the angle of rotation (the number 90). • Press ENTER on point C, the object to be rotated. Notice that the number now has a degree symbol associated with it and that the point has been rotated in the counter-clockwise direction. b^ 90* o fc 53 www.ckl2.org Step 6: What we want to do next is to rotate point B about point C through an angle of 90° in the clockwise direction. To do this we will need an angle of -90. Place this number on the screen. Using the value of -90, rotate point B about point C through an angle of -90°. Step 7: You should now have two points below the line segment BC. Use the Quad, tool to construct a quadrilateral using points B, C and the two points constructed in Steps 5 and 6. Answer Question 1 on the worksheet. Step 8: In a similar fashion, rotate point C about point A through an angle of -90° and rotate point A about point C through an angle of 90°. This will allow us to construct a second square. Use the Quad, tool again to construct the square on side AC. Step 9: Finally, rotate point B about point A through an angle of 90° and rotate point A about point B through an angle of -90°. Then construct a third square on hypotenuse AB. n c -90* Big- step 10: Start with this step if you are using the pre- constructed file "PYTHAG1". Select the Measure > Area tool and measure the area of the three squares. Step 11: Using the Calculate tool, press ENTER on the measurements of the two smaller squares and then press the + key. Place the sum off to the side of the screen. How does this sum compare to the square of the hypotenuse? Record your observations in the table for Question 2 on the worksheet. Step 12: To test your construction, drag points A, B and/or C to a new location on the screen. Answer Question 3 on the worksheet. Problem 2 — Inside a Square Proof In this problem, we are going to look at a proof of the Pythagorean Theorem. We hope to prove the statement that, if z is the length of the hypotenuse of a right triangle and x and y are the lengths of the legs of the right triangle, then z 2 = x 2 + y 2 . Step 1: Construct a line segment AB. Use the Alph-Num tool to place the value 90 on your screen. fin oE: 90 fc www.ckl2.org 54 Step 2: Access the Rotation tool and press ENTER on point A as the center of the rotation, then on 90 as the angle of rotation and finally on line segment AB as the object to be rotated. Label the new point D. Step 3: Continue by rotating line segment AD about point D through an angle of 90°. Label the new point C. Step 4: Complete the square by constructing line segment BC. Step 5: Using the Point on tool, add point E on AB as shown and overlay a line segment BE. Dp qiv: Step 6: Select the Compass tool to construct circles with radius equal to the length of BE. • Press ENTER on BE. A dashed circle will appear and follow the pointer. • Press ENTER on point A. The compass circle is anchored at center A. Create a point of intersection of this circle with AD. Label this point F. Hide the compass circle. ['? <?'- {*H- Q '■ Step 7: Use the Compass tool again to construct circles with centers at C and D and radius = BE. Create points of intersection of these circles with DC and BC. Label these points G and H. Hide the compass circles. Drag point E to confirm that F, G, and H all move as E moves. Step 8: Construct the quadrilateral EFGH. Can you prove that this quadrilateral is a square? n<i-- Step 9: Use the Alph-Num tool to place the labels x, y, and z on the figure as shown. BE is labeled x, therefore AF = DG = CH = x. 55 www.ckl2.org EA is labeled y, therefore FD = GC = HB = y. Since ABCD is a square, each of the angles at A, B, C and D are 90°, so we have four congruent triangles, namely AEFA, aFGD, aGHC and AHEB. qfi-: V eK Step 10: Start with this step if you are using the pre- constructed file "PYTHAG2". Let's examine the algebra in this situation. ABCD is a square with sides of length (x + y) . The area of the square ABCD is (x + y) 2 = x 2 + 2xy + y 2 . Each of the triangles AEFA, aFGD, aGHC and AHEB is a right triangle with height x and base y. So, the area of each triangle is -^xy. The sum of the areas of the four triangles is 4 • ^xy = 2xy. EFGH is a square with sides of length z- So the area of EFGH is z 2 . Looking at the areas in the diagram we can conclude that: ABCD = AEFA + AFGD + AGHC + AHEB + £FG# On the worksheet, substitute the area expressions (with variables x, y, and z) into the equation above and simplify. Step 11: Let's look at this numerically as well to confirm what we just proved algebraically. Measure BE, ~HB and EH. Note: Measure HB and EH by pressing ENTER on each endpoint, since these do not have separate segments constructed. Use the Calculate tool to find the squares of these lengths. Record your observations in the table for Question 6 on the worksheet. Step 12: Find the sum of the squares of the lengths of segments BE and HB. In the right triangle HBE, is BE 2 + HB 2 = EH 2 ? Drag point E to ensure that the relationship holds for other locations of the points E, F, G, and H. What would happen if you dragged one of points A or fi? Would the relationship still hold? Answers Questions 7 and 8 on the worksheet. 9.2 Investigating Special Triangles This activity is intended to supplement Geometry, Chapter 8, Lesson 4- ID: 7896 Time required: 45 minutes www.ckl2.org 56 Activity Overview In this activity, students will investigate the properties of an isosceles triangle. Then students will construct a 30° - 60° - 90° triangle and a 45° - 45° - 90° 'triangle to explore the ratios of the lengths of the sides. Topic: Right Triangles & Trigonometric Ratios • Calculate the trigonometric ratios for 45° - 45° - 90°, 60° - 60° - 60° and 30° - 60° - 90° triangles. Teacher Preparation and Notes This activity is designed to be used in a high school or middle school geometry classroom. • If needed, review or introduce the term median of a triangle. Any median of an equilateral triangle is also an altitude, angle bisector, and perpendicular bisector. • This activity is designed to be student- centered with the teacher acting as a facilitator while students work cooperatively. • The worksheet guides students through the main ideas of the activity and provides a place for students to record their work. You may wish to have the class record their answers on separate sheets of paper, or just use the questions posed to engage a class discussion. • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. • To download the calculator files, go to http://www. education, ti. com/calculators/downloads/ US/Activities/Detail?id=7896 and select EQUIL and ISOSC. Associated Materials • Student Worksheet: Investigating Special Right Triangles http://www.ckl2.org/flexr/chapter/ 9693, scroll down to the second activity. • Cabri Jr. Application . EQUIL.8xv and ISOSC.8xv Problem 1 — Investigation of Triangles First, turn on your TI-84 and press APPS. Arrow down until you see Cabri Jr and press ENTER. Open the file ISOSC. This file has a triangle with an isosceles triangle with AB = AC. Using the Perpendicular tool (ZOOM > Perp.), construct a perpendicular from point A to side BC. Label the point of intersection of this line with BC as D. To name the point, they need to select the Alph-Num tool (GRAPH > Alph-Num), select the point, and press x _1 ENTER for the letter D. Construct line segments BD and CD (WINDOW > Segment) and then measure the segments (GRAPH > Measure > D. & Length). 57 www.ckl2.org Would you have expected these segments to be equal in length? Drag point C to see the effect on the lengths of the line segments. It appears that the perpendicular from the vertex always bisects the opposite side. Measure the angles BAD and CAD. Will they always be equal? Problem 2 — Investigation of 30-60-90 Triangles Open the file EQUIL. Note that all three angles are 60° angles. Construct the perpendicular from A to side BC. Label the point of intersection as D, like in Problem 1. From the construction above, we know that D bisects BC and that mlBAD = 30°. Construct segment BD. We now have triangle BAD where mlD = 90°, mlB = 60° and mlA = 30°. We also have triangle ACD where mlA = 30°, mlC = 60° and mlD = 90°. This completes the construction of two 30° - 60° - 90° triangles. We will work only with the triangle BAD. You may choose to have the students hide the segments AC and CD. To do this, construct segments BD and AB on top of the larger triangle. Then hide the original triangle. Keep the point C. We will need that point later to resize the triangle. Measure the three sides of the triangle. Press GRAPH and select the Calculate tool. Click on the length of BD, then on the length of AB. Press the -f key. Students will see the result 0.5. Move it to the upper corner. Repeat this step to find the ratio of AD : AB and AD : BD. These ratios will become important when you start working with trigonometry. Drag point C to another location. H.90/ £.15 D i'i 50 0.B7 1.73 4.2H |t What do you notice about the three ratios? Problem 3 — Investigation of 45-45-90 Triangles Press the Y = button and select New to open a new document. To begin the construction of the 45° - 45° - 90° triangle, construct line segment AB and a perpendicular to AB at A. Use the compass tool with center A and radius AB. The circle will intersect the perpendicular line at C. www.ckl2.org 58 Hide the circle and construct segments AC and BC. Can you explain why AB = AC and why angle ACB = angle ABC? Why are these two angles 45° each? Students should notice that the two angles must be equal, and angle A is 90°. Therefore, because the sum of the angles in a triangle is 180°, the two angles must be 45° each. Measure the sides of the triangle. This verifies that AB = AC. Use the CALCULATE tool to find the ratio of AC : BC and AC : AB. Once again, these ratios will be important when you study trigonometry. :-:.bm ^ 0.?i i.Od \ SiB" J.BO Drag point B and observe what happens to the sides and ratios. Why do the ratios remain constant while the sides change? Students should notice that AC and AB are equal, so the ratios will always remain the same. 9.3 Ratios of Right Triangles This activity is intended to supplement Geometry, Chapter 8, Lesson 5. ID: 11576 Time Required: 45 minutes Activity Overview In this activity, students will explore the ratios of right triangles. Students will discover that they can find the measure of the angles of a right triangle given the length of any two sides. Topic: Right Triangles & Trigonometric Ratios • Sine • Cosine • Tangent Teacher Preparation and Notes 59 www.ckl2.org • This activity was written to be explored on the TI-84 with the Cabri Jr. and Learning Check appli- cations. • Before beginning this activity, make sure that all students have the Cabri Jr. applications. Also, make sure students have or know the trigonometric definitions. • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. • To download the calculator file, go to http://www. education, ti . com/calculators/downloads/ US/ 'Activities/Detail? id=11576 and select TRIG. Associated Materials Student Worksheet: Ratios of Right Triangles http://www.ckl2.org/flexr/chapter/9693, scroll down to the third activity. Cabri Jr. Application TRIG.8xv Problem 1 — Exploring Right Triangle Trigonometry You may need to allow students to use a textbook (or other resource) to hnd the definitions of sine, cosine, and tangent. Students are asked to give the ratio of several triangles on their handheld or their accompanying worksheet. Problem 2 — Exploring the Sine Ratio of a Right Triangle For this problem, students will investigate the sine ratio of two sides of a triangle. Students should start the Cabri Jr. app and open the file Trig.8xv. Students will collect data on their worksheets by moving point B. They will do this for four different positions of the point. ME $.2 EC £.2 ML 1.7 Htfc: BtYAB O.HE hlYAB ij,9i BC/hC if.H? Students will discover that the ratio of BC to AB remains constant, no matter how large the triangle is; Therefore, students will be able to use the inverse of sine to find the measure of the angles in AABC. Students will need to answer several questions on their handhelds or their accompanying worksheets. Problem 3 — Exploring the Cosine Ratio of a Right Triangle Students will repeat the exploration in Problem 2, but with the cosine ratio. www.ckl2.org 60 Problem 4 — Applying the Sine, Cosine, and Tangent Ratio of a Right Triangle In Problem 4, students are asked to apply what they have learned about how to find the measure of an angle of a right triangle given two sides of the triangle. £in-K2/5) .4115168461 sin-K2/5> 23.57817843 cos-K2/"5) 66.42182152 Solutions 1. For right triangle ABC, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. 2. For right triangle ABC, the cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. 3. For right triangle ABC, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. 4-1 M M M 9. | 10. Sample answers: Table 9.1: Position BC AB BC AB sin ■1 BC AB 1 2 3 4 2.4376781463393 3.0769811671077 3.6665092204124 4.3154341679767 6.2006451991814 7.8268201774092 9.3263841370716 10.977034107736 0.39313298342907 0.39313298342907 0.39313298342906 0.39313298342905 23.149583787224 23.149583787224 23.149583787223 23.149583787222 11. The ratio does not change. 12. No, the angle does not change. 13. 23.1496 14. 66.8504 15. Sample answers: 61 www.ckl2.org Table 9.2: Position AC AB % cos" 1 % 1 7.0816099136391 8.8549125969341 0.7997379800328 36.894911430193 2 8.0238624186986 10.033114118664 0.79973798003277 36.894911430196 3 9.0235078139592 11.283080257848 0.79973798003279 36.894911430194 4 3.7704816328074 4.7146462053143 0.79973798003281 36.894911430192 16. 36.8949 17. 53.1051 18. A = tan" 1 f§ 19. A = 23.57,5 = 66.42 20. A = 21.8,5 = 68.2 21. A = 23.96,5 = 66.04 22. A = 53.13,5 = 36.87 23. A = 15.07,5 = 74.93 24. A = 42.83,5 = 47.17 25. A = 45,5 = 45 26. A = 29.05,5 = 60.95 www.ckl2.org 62 Chapter 10 TE Circles - TI 10.1 Chords and Circles This activity is intended to supplement Geometry, Chapter 9, Lesson 3. ID: 9773 Time required: 30 minutes Topic: Circles • Deduce from the Perpendicular Bisector Theorem the following corollaries: a) The perpendicular from the center of a circle to a chord bisects the chord. b) The line joining the center of a circle to the midpoint of a chord is perpendicular to the chord. c) The center of a circle is at the intersection of the perpendicular bisector of two non-parallel chords. Activity Overview Students will begin this activity by exploring how the chord in a circle is related to its perpendicular bisector. Investigation will include measuring lengths and distances from the center of the circle. These measurements will then be transferred to a graph to see the locus of the intersection point of the measurements as the endpoint of a chord is moved around the circle. In the extension, students will be asked to find an equation for the ellipse that models the relationship. Teacher Preparation This activity is designed to be used in a high school geometry classroom. • Students should already be familiar with circles, chords of circles, and perpendicular bisectors. Classroom Management • This activity is designed to be student- centered with the teacher acting as a facilitator while students work cooperatively. Use the following pages as a framework as to how the activity will progress. • Information for an optional extension is provided at the end of this activity. 63 www.ckl2.org • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. Associated Materials • Student Worksheet: Chords and Circles http://www.ckl2.org/flexr/chapter/9694 • Cabri Jr. Application Introduction When hikers and skiers go into terrain where there is a risk of avalanches, they take safety equipment including avalanche rescue beacons. An avalanche rescue beacon sends and receives electromagnetic field signals that can travel up to about 30 meters. The search pattern used to locate a beacon buried in the snow is based on the properties of chords and diameters in circles. In this activity, you will use Cabri Jr. to model an avalanche search pattern. Problem 1 — Relationship between a chord an its perpendicular bisector Construct a circle using the Circle (F2 >Circle) too in Cabri Jr. to represent the beacon signal. Use the Hide/Show to hide its center. Construct a chord to represent the path of a rescuer as he walks a straight path until the signal disappears. Use the Segment tool to draw the chord with endpoints on the circle and label it. The rescuer walks back to the midpoint of this path. Find the midpoint of AB and label it M. Construct a line perpendicular to AB through M, to represent the rescuer walking away from the path at a 90° angle until the signal disappears. Find one intersection point of the perpendicular line and the circle. Label it C. The rescuer turns around and walks in the opposite direction until the signal disappears again. Find the other intersection point of the perpendicular line and the circle. Label it D. Hide the perpendicular line. Construct a segment connecting points C and D. The rescuer walks back to the midpoint of this new path. Find the midpoint of CD and label it X. This will be the center of the circle formed by the beacon signals. Dig for the missing person! Confirm that you have located the center of the circle. Measure the distances from X to A, B, C, and D. www.ckl2.org 64 Problem 2 — Extension Extension 1 Write a proof of the relationship used in the activity. Given a chord of a circle and its perpendicular bisector, prove that the perpendicular bisector passes through the center of the circle. Extension 2 Use a compass and straightedge to construct a circle and a chord. Construct the perpendicular bisector of the chord and see that it passes through the center of the circle. 10.2 Inscribed Angles Theorem This activity is intended to supplement Geometry, Chapter 9, Lesson 4- ID: 12437 Time Required: 15 minutes Activity Overview Students will begin this activity by looking at inscribed angles and central angles and work towards discovering a relationship among the two, the Inscribed Angle Theorem. Then, students will look at two corollaries to the theorem. Topic: Circles • Construct central and inscribed angles • Inscribed angles theorem Teacher Preparation and Notes • This activity was written to be explored with the Cabri Jr. application on the TI-84- • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. • To download the calculator files , go to http: //www . education, ti. com/calculators/downloads/ US/Activities/Detail?id=12437 and select INSCRIB1-INSCRIB5. Associated Materials • Student Worksheet: Inscribed Angle Theorem http://www.ckl2.org/flexr/chapter/9694, scroll down to the second activity. • Cabri Jr. Application . INSCRIB1.8xv, INSCRIB2.8xv, INSCRIB3.8xv, INSCRIB4.8xv, and INSCRIB5.8xv Problem 1 — Similar Triangles Students will begin this activity by looking at inscribed angles and central angles and work towards discovering a relationship among the two. 65 www.ckl2.org Students will be asked to collect data by moving points A and C. Students are asked questions about the relationships in the circle and are asked to make a conjecture. In order to calculate the ratio of mlACB to mlADB, students can use the Calculate tool in the Actions menu. Q -* M jf\ * .■fc£i *J* ■ft angled 73.30* On page 1.6, students will look at two inscribed angles intercepted by the same arc and are asked to make a conjecture about the relationship. In an advanced setting a proof of the inscribed angle theorem and the two conjectures in problem one are appropriate and can be proved using isosceles triangles. Problem 2 — Extension of the Inscribed Angle Theorem In Problem 2, students will look at two more angles created from the central angle and the intercepted arc. Both sections of this problem are corollaries of the Inscribed Angle Theorem and both solutions are congruent to the measure of the central angle intercepted by the arc or one-half the measure of the central angle. f Jr V* a lie ^ — """ FinGLEtefi.fiS* priijLE['E,3. sir- Solutions 1. Sample answers Table 10.1: Position Measure of lACB Measure of lADB mlACB mlADB 1 2 3 4 36.65° 49.50° 49.50° 49.50° 73.30° 99.01° 99.01° 99.01° 0.5 0.5 0.5 0.5 2 i 3. Sample answers www.ckl2.org 66 Table 10.2: Position Measure of LACB Measure of lAEB 1 2 3 4 36.65° 48.30° 48.30° 48.30° 36.65° 48.30° 48.30° 48.30° 4. Sample answer: They are congruent. 5. diameter of the circle 6. 90° 7. Sample answers Table 10.3: Position Measure of LACB Measure of lADB Measure of lAGE 1 2 3 4 36.65° 67.04° 67.04° 67.04° 73.30° 134.10° 134.10° 134.10° 36.65° 67.04° 67.04° 67.04° 9. Sample answers Table 10.4: Position Measure of LACB Measure of LADB Measure of LABE 1 2 3 4 57.65° 49.20° 54.90° 63.55° 115.30° 98.40° 109.80° 127.10° 57.64° 49.19° 54.90° 63.54° 10. 10.3 Circle Product Theorems This activity is intended to supplement Geometry, Chapter 9, Lesson 6. ID: 12512 Time Required: 20 minutes Activity Overview Students will use dynamic models to find patterns. These patterns are the Chord- Chord, Secant-Secant, and Secant- Tangent Theorems. 67 www.ckl2.org Topic: Circles • Chord- Chord, Secant- Secant, and the Secant- Tangent Product Theorems Teacher Preparation and Notes • This activity was written to be explored with the Cabri Jr. application on the TI-84- • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. • To download the calculator files , go to http://www. education, ti. com/calculators/downloads/ US/Activities/Detail?id=12512 and select PRODUC1-PRODUC5. Associated Materials • Student Worksheet: Circle Product Theorems http://www.ckl2.org/flexr/chapter/9694, scroll down to the third activity. • Cabri Jr. Application . PR0DUC1.8xv, PRODUC2.8xv, and PRODUC3.8xv Problem 1 — Chord-Chord Product Theorem Students will begin this activity by investigating the intersection of two chords and the product of the length of the segments of one chord and the product of the length of the segments of the other chord. Students will be asked to collect data by moving point A. Students are asked to calculate the products by hand on their accompanying worksheet. Students are asked several questions about the relationship among the products. As an extension, prove the chord-chord product theorem using similar triangles. J^- r r i ^E * f ---■^ DO 5 \ AK2.70 ^-_ ^j SKG.SS CKi.Ei DK i.7B Problem 2 — Secant-Secant Product Theorem Students will investigate the intersection of two secants and the product of the lengths of one secant segment and its external segment and the product of the lengths of the other secant segment and its external segment. Students will be asked to collect data by moving point A. They are to calculate the products by hand on their accompanying worksheet. Students are asked several questions about the relationship among the products. As an extension, prove the secant-secant product theorem using similar triangles. www.ckl2.org 68 f r- ■-T-- fiK E.9H C« 6.65 DK 3. 42 Problem 3 — Secant-Tangent Product Theorem Students will investigate the intersection of the product of the lengths of one secant segment and its external segment and the square of the tangent segment. Students will be asked to collect data by moving point A. Students are asked to calculate the products by hand on their accompanying worksheet. Students are asked several questions about the relationship among the products. As an extension, prove the secant-tangent product theorem using similar triangles. ( CK fi.13 M 2.90 Problem 4 — Application of the Product Theorems Students will be asked to apply what they learned in Problems 1-3 to solve a few problems. Solutions 1. Sample answers: Table 10.5: Position AX BX cx DX AX- BX CXDX 1 2.70 0.99 1.51 1.78 2.67 2.68 2 2.51 1.05 1.38 1.90 2.63 2.62 3 2.93 0.91 1.80 1.49 2.67 2.68 4 1.80 1.08 2.51 0.77 1.94 1.93 2. They are equal. 3. equal 4. Sample answers: 69 www.ckl2.org Table 10.6: Position AX BX CX DX AXBX CXDX 1 5.94 3.82 6.65 3.42 22.69 22.74 2 4.85 3.16 5.85 2.62 15.33 15.33 3 4.03 2.75 5.32 2.09 11.08 11.12 4 7.47 4.96 7.92 4.68 37.05 37.07 5. They are equal. 6. equals 7. Sample answers: Table 10.7: Position AX CX DX AX 2 CXDX 1 4.22 6.13 2.90 17.81 17.77 2 3.42 5.40 2.16 11.70 11.66 3 2.45 4.56 1.32 6.00 6.02 4 9.46 11.21 7.98 89.49 89.46 8. They are equal. 9. equals 10. 6 11 2 ±±. 4 12. 3Vl3or 10.817 www.ckl2.org TO Chapter 11 TE Perimeter and Area - TI 11.1 Diameter and Circumference of a Circle This activity is intended to supplement Geometry, Chapter 10, Lesson 4- ID: 9844 Time required: 30 minutes Topic: Circles • Use technology to verify the circumference and area formulas for the circle. Activity Overview In this activity, students explore the relationship between a circle's circumference and its diameter. This will lead students to their own discovery of a value for pi. Teacher Preparation This activity is designed to be used in a high school geometry clssroom. • Students should already be familiar with circles, diameter, circumference, and pi. Classroom Management • This activity is designed to be student- centered with the teacher acting as a facilitator while students work cooperatively. Use the following pages as a framework as to how the activity will progress. • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail?id=258#. Associated Materials • Student Worksheet: Diameter and Circumference of a Circle http : //www . ckl2 . org/f lexr/chapter/ 9695 • Cabri Jr. Application 71 www.ckl2.org In this activity we will • Draw a circle • Measure the diameter of the circle • Measure the circumference of the circle • Calculate the ratio of the circumference to the diameter. Problem 1 Press APPS. Move down to the Cabri Jr. APP and press ENTER. Press ENTER, or any key, to begin using the application. Press Y = for the F\ menu and select New. (If asked to Save changes? press <— ENTER to choose "No.") Press WINDOW for the F2 menu, move down to Circle, and press ENTER. Press ENTER to mark the center of the circle, then move the pencil to indicate the length of the radius, and press ENTER to complete the circle. Draw a line through the two points which determined the circle. To do this, press WINDOW for the F2 menu, move to Line, then press ENTER. Move the pencil until the point on the circle is flashing, and press ENTER. Now move the pencil until the center of the circle is flashing, and press ENTER. Press CLEAR to exit the line drawing tool. \. 4 k / Press WINDOW for F2 and move to Point. Move to the right and down to select Intersection. Press ENTER. Move the pencil until both the line and the circle are flashing. Press ENTER to mark the point which is the intersection of the circle and the line. Now we have two points on the circle which are the endpoints of a diameter To measure the circle's diameter, press GRAPH for F5 and move down and right to select Measure, D. &: Length. Press ENTER. Move the pencil until one endpoint of the diameter is flashing then press ENTER. Move to the other endpoint of the diameter and when it is flashing, press ENTER. Press + to see the measurement rounded to hundredths. The hand is active so you can move the measurement to a convenient location then press ENTER. The Measurement tool is still active so now you can find the circumference of the circle. Move the pencil until the circle is flashing. Press ENTER then + to see the circumference rounded to hundredths. Move the hand until the measurement is in a convenient location. Press ENTER. Press CLEAR to turn off the measurement tool. Press GRAPH for F5 and move down to Calculate. Press ENTER. Move the arrow until the circum- ference measurement shows a flashing underline and press ENTER then -K Move the arrow until the diameter measurement has a flashing underline and press ENTER again. The number displayed is the ratio of the circle's circumference to its diameter. www.ckl2.org 72 To explore this relationship with other circles, press CLEAR to turn off the Calculate tool. Move the arrow until the point which defined the circle's radius or its center is flashing. Press ALPHA to activate the hand. Grab the point and move it to change the size of the circle. To confirm that the ratio is still 3.14, repeat the Calculate procedure. (It is actually being recalculated each time the circle changes, but it is impossible to tell this since the number is unchanging.) To exit the APP, press Y = for the Fl menu. Move to Quit, then press ENTER. 11.2 From the Center of the Polygon This activity is intended to supplement Geometry, Chapter 10, Lesson 6. ID: 11644 Time Required: 45 minutes Activity Overview Students will explore the area of a regular polygon in terms of the apothem and the perimeter. They will derive the formula for a regular pentagon and regular hexagon. Then, students will see how the formula relates to the formula for the area of triangles. Students will then be asked to apply what they have learned about the area of a regular polygon. Topic: Quadrilaterals & General Polygons • Regular Polygons • Area of Regular Polygons Teacher Preparation and Notes • This activity was written to be explored with the Cabri Jr. app on the TI-84- • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. • To download the calculator files , go to http: //www . education, ti. com/calculators/downloads/ US/Activities/Detail?id=11644 and select PENTAGON, HEXAGON, OCTAGON. Associated Materials • Student Worksheet: From the Center of a Polygon http://www.ckl2.org/flexr/chapter/9695, scroll down to the second activity. • Cabri Jr. Application . PENTAG0N.8xv, HEXAG0N.8xv, 0CTAG0N.8xv Problem 1 — Area of a Regular Pentagon Students will begin this activity by looking at a regular pentagon. In file PENTAGON, students are given regular pentagon ABCDE with center 7?. Students are given the length of CD (side of the polygon), RM (apothem), and the area of the polygon. Students are to collect data in the table given on their student worksheet and contains 4 columns: Apothem, Perimeter, a ■ p (apothem times perimeter), and Area. 73 www.ckl2.org Students will collect data by moving point D. They will do this for four different positions of the point. Students are asked about how the area and the apothem times the perimeter are related. F:Mi. 72 .j K CD ZSO HF:Ert i0.r=: t ~~1 ^ E * \ / \ / cb— Am-iJ[i Problem 2 — Area of a Regular Hexagon Students will repeat the same process as Problem 1 for a regular hexagon. Students will begin to discover the formula for a regular polygon is one-half the perimeter times the apothem. Problem 3 — Area of a Regular Polygon In this problem, students are to "prove" the formula for the area of a regular polygon by looking at an octagon and the triangles created by the radii of the octagon. If students are confused by the term radius of the polygon, explain that this is the radius of a circle circumscribed about a regular polygon. Problem 4 — Area of Regular Polygons In Problem 4, students are asked to apply what they have learned about the area of regular polygons The students are given a question on the area of a regular polygon and a calculator on each page. The students are to use the calculator to find the area. Solutions 1. Sample answers: Table 11.1: Position Apothem (a) Perimeter (P) a ■ p times ter) (apothem perime- Area 1 1.72 12.5 21.5 10.75 2 1.93 14 27.02 13.49 3 2.13 15.5 33.015 16.53 4 2.55 18.5 47.175 23.55 2. area = ^ (apothem) (perimeter) 3. Sample answers: \ap www.ckl2.org 74 Table 11.2: Position Apothem (a) Perimeter (p) a • p times ter) (apothem perime- Area 1 1.91 13.2 25.2 12.6 2 2.25 15.6 35.1 17.76 3 2.34 16.2 37.908 18.9 4 2.60 18 46.8 23.4 4. Area = I (apothem) (perimeter) = ^ap 5. 10 6. Yes 7. Yes 8. Area CDR = -as 9. Area = 2 as (&) = ^ as 10. Area = -^asn 11. 485.52 sq. in. 12. 25.2 sq. cm 13. 501.84 sq. ft 14. 1,039.2 sq. mm T5 www.ckl2.org Chapter 12 TE Surface Area and Volume TI 12.1 Surface Area of a Cylinder This activity is intended to supplement Geometry, Chapter 11, Lesson 2. ID: 10075 Time required: 30 minutes Activity Overview Students will explore a net representation for a right cylinder. The surface area will be developed from the parts of the net. Topic: 3-Dimensional Geometry • Construct 3- dimensional prisms and pyramids from nets. • Calculate the surface area of a right prism or cylinder. Teacher Preparation and Notes • This activity is designed to be used in a high school or middle school geometry classroom. • This activity is designed to be student- centered. • The surface area of a right cylinder with base radius = R and height = H is SA = 2nR 2 + 2nRH. The activity asks students to notice that the circumference of the circle is the length of the rectangle in the net. • The points R and H control the radius and the height of the cylinder. When R is dragged, the length of the rectangle also changes (because the length = circumference of the circle). The height of the rectangle does not change when R is dragged. • Note: Measurements can display 0, 1, or 2 decimal digits. If digits are displayed, the value shown will round from the actual value. To change the number of digits displayed: 1. Move the cursor over the value so it is highlighted. 2. Press + to display additional decimal digits or - to decrease digits. www.ckl2.org T6 • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail?id=258#. • To download the calculator file, go to http://www. education, ti . com/calculators/downloads/ US/Activities/Detail?id=10075 and select CYLINDER. Associated Materials • Student Worksheet: Surface Area of Cylinders http://www.ckl2.org/flexr/chapter/9696 • Cabri Jr. Application . CYLINDER. 8xv Problem 1 — Nets A net is a pattern that can be cut out and folded into a 3- dimensional figure. Students should see a partial net of a right cylinder. If the rectangle of the net were rolled up, the circle would be the top face of the cylinder (like a lid of a jar). ?•%— °— f i.oo Ah i.PO The dimensions of the net can be changed by dragging the points R and H. Students should drag these points and notice what changes with the figure for each point. When R is moved, the radius of the circle and the width of the rectangle are changed. Students should use the D. & Length tool (F5 > Measure) to find the length of the rectangle. Note: Display measurements with 2 decimal digits. To do this, hover the cursor over the measurement and then press the plus key (+). Next, they should use the Calculate tool to divide the length of the rectangle by the radius of the circle to find that the width of the rectangle is the same as the circumference of the circle. Problem 2 — Surface Area Students should use the Area tool from the Measure menu (F5 > Measure) to find the areas of the rectangle and the circle. Then, they should use the Alph-Num tool to place the number 2 on the screen. Note: Press the ALPHA button to access numerical characters. The tool icon in the corner of the screen will display A . Press the ENTER button to start and end the text. ,..-—-■, :=.iH i R^— °— $ ij)0 S.2H i H i 2. SO i iE.ro II www.ckl2.org Finally, students are to find the surface area of the cylinder. Remind students that this is only a partial net, so one of the faces is missing. To find the surface area, students need to find the area of the circle and the area of the rectangle. They should first use the Area tool from the Measurement menu to calculate these areas. Then, they need to use the Calculate tool to find the sum of both circle bases by clicking on the area of the circle, pressing I, and clicking on the 2. Next, they should click on the number they just calculated, press +, and click on the area of the rectangle. www.ckl2.org 78 Chapter 13 TE Transformations - TI 13.1 Transformations with Lists This activity is intended to supplement Geometry, Chapter 12, Lessons 2 and 3. ID: 10277 Time required: 40 minutes Activity Overview Students will graph a figure in the coordinate plane. They will use list operations to perform reflections, rotations, translations and dilations on the figure, and graph the resulting image using a scatter plot. Topic: Transformational Geometry • Perform reflections, rotations, translations and dilations using lists and scatter plots to represent figures on a coordinate plane. Teacher Preparation and Notes This activity is designed to be used in a high school geometry or algebra classroom. If an original point on the coordinate plane is denoted by (x,y), then each of the following ordered pairs denotes a transformation: (x, —y) reflect over x — axis (— x,y) reflect over y — axis (y, x) reflect over y = x (— y,x) rotate 90° around origin {—x, —y) rotate 180° around origin (y, -x) rotate 90° around origin To perform a translation, add or subtract a constant from the list with the x— values or the y— values of the figure. To perform a dilation, multiply a constant scale factor by the list with the x- values or the y- values of the figure. This activity is designed to be student- centered with the teacher acting as a facilitator while students work cooperatively. If desired, have students work in groups of 3. Each person in the group should enter a different combination of lists for Problem 2 and the group should discuss the results. 79 www.ckl2.org Associated Materials Student Worksheet: Translations with Lists http://www.ckl2.org/flexr/chapter/9697 Problem 1 — Creating a Scatter Plot Before beginning the activity, students need to clear all entries from the Y = screen and all lists. First, students will enter the data on the worksheet into lists L\ and L2. After setting up Plotl for a scatter plot of L\ vs L2 and changing the window settings, students view the graph and sketch the figure on the worksheet. The shape should be an arrow, in the first quadrant, pointing to the right. Problem 2 — Reflections and Rotations In the list editor students are to enter the formulas = —L\ and = — L-i for L3 and L4 respectively. This will allow them to create several different reflections and rotations of the original figure. To type Li, students need to press 2 [1]. To type L2, students need to press 2 nd [2]. For each combination of lists, students are to determine what type of reflection occurred. A: x <— L3 and v <— L2 ...<£ k> (— x,y) over y-axis B: x <— L\ and y <— L4 k?> i$ (x, —y) over x-axis C: x <— L2 and y <— L\ www.ckl2.org 80 M> (y, x) over the line y = x Use Plot2 to create the following scatter plots. For each combination, determine what type of rotation occurred. D: x <— Li and y <— L\ tr : ^> (— y, x) 90° around origin E: x <— L2 and y <— L3 h> : '"••■-■•■"" (-x, -y) 90° around origin F: x <— L3 and y <— L4 fe>... <£ (y, -x) 180° around origin Problem 3 — Translations In the list editor, students are to enter the formulas = L\ — 5 and = L2 + 3 for L3 and L4 respectively. This will allow them to translate the original figure. Students should see that the image shifted to the left 5 units and up 3 units. Remind students that the 81 www.cki2.0rg tick marks on the graph are every 2 units. L £>... Now students are to translate the scatter plot into Quadrant 3 by editing the formula bars for L3 and L4. Possible formulas are below. L3 formula: = L\ — 15 The image shifted 15 units to the left and 10 units down. L4 formula: = hi — 10 te> & Extension — Dilations Remind students that they have seen dilations in Chapter 7, Lesson 6. In the list editor, students are to enter the formulas = 0.5*Li and = 0.5*L2 for L3 and L4 respectively. This will allow them to dilate the original figure. Students should see that the image decreased in size. If they have trouble seeing the image, they can the mark of the plot to the small dot. Then students are to dilate the scatter plot into Quadrant 3 by editing the formula bars for L3 and L4. Remind students that the scale factor needs to be the same for both lists. Possible formulas are below. L3 formula: = -0.5*Li L4 formula: = -0.5*L2 13.2 Reflections and Rotations This activity is intended to supplement Geometry, Chapter 12, Lesson 5. www.ckl2.org 82 ID: 8180 Time required: 45 minutes Activity Overview Students draw a puppet whose arms and legs are constructed by using a translation vector and reflection line. They will manipulate the vector and observe the changes in the puppet. This activity will help students visualize the effect of a transformation in their construction. Topic: Transformational Geometry • Given a translation vector and a geometric figure, translate the figure to discover that lengths, angles, areas, shapes and parallel line segments are preserved under translations. • Given a reflection line and a geometric figure, reflect the figure to discover that lengths, angles, areas and shapes are preserved under reflections and orientations are reversed. Teacher Preparation and Notes • Although this is a geometry activity, it can also be used in a Pre-Algebra or Algebra classroom. • Before beginning this activity, students should be familiar with reflections and translations outside the coordinate plane. They should already know the terms associated with transformations, such as line of reflection and translation vector. • This activity is designed to be explored individually or in pairs, with the teacher acting as a facilitator. • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. Associated Materials • Student Worksheet: Reflections <fe Rotations http://www.ckl2.org/flexr/chapter/9697, scroll down to the second activity. • Cabri Jr. Application Problem 1 — Constructing the puppet In Cabri Jr., students will start with a new document. They will use the Circle, Segment, and Point On tools to create the head, body, and shoulder, respec- tively. Students must draw the segment for the translation vector in the order described on the worksheet (bottom right point, then top left point). 83 www.ckl2.org They are directed to use the Translation tool to create the left hand and then use the Reflection tool to create the right hand. Each point will need to be joined with a segment to the shoulder point. Another segment to be used as the translation vector for the legs will be drawn with the first one. Again, students must draw the segment in the order described on the worksheet (top right point, then bottom left point). Problem 2— Moving the puppet Students should drag the point joining the two translation vectors and observe the changes of the arms and legs. J> Ni To investigate the puppet further, students should use the Slope tool to find the slopes of the arms. Then they can drag the point on the left side of the screen and observe the changes. Students can also find the coordinates of the left and right hands. ^ y\, 13.3 Perspective Drawing This activity can be used as an end of the year activity or to supplement the "Know What?" from Chapter 7, Lesson 6. ID: 10033 Time required: 60 minutes Activity Overview In this activity, students draw figures in one- and two-point perspective and compare and contrast the two types of drawings. They then create an isometric drawing and compare it to their drawings in perspective. Topic: 3— Dimensional Geometry • Construct 3- dimensional prisms and pyramids. • Record the number of faces, edges, and vertices of prisms. Teacher Preparation and Notes www.ckl2.org 84 • Perspective drawings can be taught at any time in a geometry curriculum, but are most appropriate after lessons on parallel and perpendicular lines, three-dimensional figures, and symmetry. • Throughout the activity, students use many drawing and construction tools, such as the Segment, Parallel, and Perpendicular tools. In this document, the first use of a tool is by name and accom- panied by its location within the menu structure. For subsequent uses, the tool may be mentioned by name or its function, and the menu location is omitted. • This activity is designed to have students explore individually or in pairs. • To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/ Software/Detail? id=258#. • To download the calculator files, go to http: //www . education, ti. com/calculators/downloads/ US/Activities/Detail?id=10033 and select RECPRISM, TRIPRISM, TWOPERSP, ISO- DRAW. Associated Materials • Student Worksheet: Perspective Drawing http : //www . ckl2 . org/f lexr/chapter/9697, scroll down to the third activity. • Cabri Jr. Application . RECPRISM.8xv, TRIPRISM.8xv, TWOPERSP.8xv, and ISODRAW.8xv Problem 1 — One-point perspective Students will open the Cabri Jr. file RECPRISM and find a vaishing point A along the horizon and a rectangle BCDE. They are to use the Segment tool from the F2 menu to draw AB and AC. These two segments (and any other segment that joins a vertex of the prism to the vanishing point) are hereafter referred to as vanishing segments. Next, they will create FG such that F is between A and B, G is between A and C, and FG x BC (Refer to the diagram at the right). To do this, they will need to do the following: . Place a point F on AB. (F2 > Point > Point On) Construct a line through F parallel to BC. (F3 > Parallel) • Plot point G at the intersection of the parallel line and AC. (F2 > Point > Intersection) • Hide the line. (F5 > Hide/Show > Object) • Draw FG. Students can now hide the vanishing segments, AB and AC, and draw BF and CG. The front and top faces of the rectangular prism have been constructed. Ask students if they are satisfied that this drawing completely represents a rectangular prism. Have them drag point A (press APLHA to grab the point) to either the far left or far right and ask students what is wrong with the drawing. (The rectangular prism does not have a face on either side.) To complete the prism, students next need to construct AD and AE. Then, using the Perpendicular tool (F3 > Perp.), construct two lines perpendicular to FG — one through F and the other through G. The 85 www.ckl2.org vertical "back" edges GH and FJ may then be created by plotting the intersection points, hiding the lines, and constructing the segments as needed. To show all six edges, students should draw a segment connecting H and /. Once all of the vanishing segments are hidden, students will find they need to also construct segments DH and EJ. H -a- & 1 Depending on the topics you have recently covered in class, you can now have students identify parallel, intersecting, and skew lines; count faces, vertices, and edges; or discuss the similarity of rectangles BCDE and FGHJ. When students have finished exploring their prisms, they may wish to hide or dash the edges that would not be visible if the figure were not transparent. The appearance of the edges may be altered using the Display tool (F5 > Display). Students are to open TRIPRISM and create a triangular prism in one-point perspective on their own. Be sure to circulate around the room and assist students as needed. Allow students a few minutes to explore the figure by dragging point A along the horizon or dragging an edge. Vertices B, C, and D can also be dragged on this figure. Problem 2 — Two-point perspective The Cabri Jr. file TWOPERSP contains two vanishing points, A and B, signifying this rectangular prism will be drawn in two-point perspective. The file also contains a vertical line segment, CD, which will be the front edge of the prism. CD is constructed so that dragging point C changes the segment's length and dragging point D translates the segment. Next, students will then use the Segment tool to draw the vanishing segments AC, AD, BC, and BD. Then they should use the Point On, Parallel, Intersection, Hide/Show, and Segment tools (as they did in Problem 1) to construct the vertical edges, EF and GH, of the left and right faces. The top and bottom edges of those faces, CE, CG, DF, and DH may be drawn as well. Students should now hide the four vanishing segments, and then draw four more: from E and F to B and from G and H to A. The intersection of the upper two vanishing segments and that of the lower two vanishing segments should be plotted and a segment drawn between them to form the last vertical edge, JK. www.ckl2.org 86 The vanishing segments may then be hidden so that students can draw the remaining edges of the prism. As before, allow students some time to drag edges, vertices, and vanishing points, observing how the prism changes as they do so. (Dragging CD is very interesting!) Again, students can either hide or dash any "hidden edges." Problem 3 — An isometric drawing Using ISODRAW, students will construct an isometric drawing of a rectangular prism. The three seg- ments shown represent the three "front" edges of the prism, and each of the angles the segments form measures 120°. Students should first use the Parallel, Intersection, Hide/Show, and Segment tools to construct AE, CF , DE, and DF, as shown in the diagram to the right. They may then use the same sequence of tools once more to construct the remaining edges: AG, CG, EH, FH, and GH. I "7i After completing the isometric drawing, students should compare it to the perspective drawings from Problems 1 and 2. As a similarity, students may identify that right angles "in real-life" do not actually measure 90° in the drawings. As a difference, all parallel lines "in real-life" are parallel in an isometric drawing, whereas in perspective drawings, some parallel lines actually "meet" at the vanishing point. 87 www.ckl2.org